Yes. You can do it by using AdminLinkProviderForUser https://docs.aws.amazon.com/cognito-user-identity-pools/latest/APIReference/API_AdminLinkProviderForUser.html

The idea is:

1. In PreSignUp lambda hook, we Link Provider to User if User already signed up. E.g:

import CognitoIdentityServiceProvider from 'aws-sdk/clients/cognitoidentityserviceprovider'

const cognitoIdp = new CognitoIdentityServiceProvider()
const getUserByEmail = async (userPoolId, email) => {
 const params = {
   UserPoolId: userPoolId,
   Filter: `email = "${email}"`
 }
 return cognitoIdp.listUsers(params).promise()
}

const linkProviderToUser = async (username, userPoolId, providerName, providerUserId) => {
 const params = {
   DestinationUser: {
     ProviderAttributeValue: username,
     ProviderName: 'Cognito'
   },
   SourceUser: {
     ProviderAttributeName: 'Cognito_Subject',
     ProviderAttributeValue: providerUserId,
     ProviderName: providerName
   },
   UserPoolId: userPoolId
 }

 const result = await (new Promise((resolve, reject) => {
   cognitoIdp.adminLinkProviderForUser(params, (err, data) => {
     if (err) {
       reject(err)
       return
     }
     resolve(data)
   })
 }))

 return result
}

exports.handler = async (event, context, callback) => {
 if (event.triggerSource === 'PreSignUp_ExternalProvider') {
   const userRs = await getUserByEmail(event.userPoolId, event.request.userAttributes.email)
   if (userRs && userRs.Users.length > 0) {
     const [ providerName, providerUserId ] = event.userName.split('_') // event userName example: "Facebook_12324325436"
     await linkProviderToUser(userRs.Users[0].Username, event.userPoolId, providerName, providerUserId)
   } else {
     console.log('user not found, skip.')
   }

 }
 return callback(null, event)
}

2. Then when user use OAuth with Facebook/Google with User Pool, the Pool will return this User linked.

Note: You may see 2 records in User Pool UI, but when access User record detail, They already merged.

The solution I created handles, I think, all cases. It also tackles some common issues with Cognito.

• If the user is signing up with an external provider, link them to any existing account, including Cognito (username/password) or external provider account.
• When linking to existing accounts, link only to the oldest account. This is important is you have more than 2 login options.
• If the user is signing up with Cognito (username/password), if an external provider already exists, reject the signup with a custom error message (because the accounts cannot be linked).

Note that when linking accounts, the Cognito pre-signup trigger returns an "Already found an entry for username" error. Your client should handle this and reattempt authentication, or ask the user to sign in again. More info on this here:

Cognito auth flow fails with "Already found an entry for username Facebook_10155611263153532"

Here is my lambda, executed on the Cognito pre-signup trigger

const AWS = require("aws-sdk");
const cognito = new AWS.CognitoIdentityServiceProvider();

exports.handler = (event, context, callback) => {

  function checkForExistingUsers(event, linkToExistingUser) {

    console.log("Executing checkForExistingUsers");

    var params = {
      UserPoolId: event.userPoolId,
      AttributesToGet: ['sub', 'email'],
      Filter: "email = \"" + event.request.userAttributes.email + "\""
    };

    return new Promise((resolve, reject) =>
      cognito.listUsers(params, (err, result) => {
        if (err) {
          reject(err);
          return;
        }
        if (result && result.Users && result.Users[0] && result.Users[0].Username && linkToExistingUser) {
          console.log("Found existing users: ", result.Users);
          if (result.Users.length > 1){
            result.Users.sort((a, b) => (a.UserCreateDate > b.UserCreateDate) ? 1 : -1);
            console.log("Found more than one existing users. Ordered by createdDate: ", result.Users);
          }
          linkUser(result.Users[0].Username, event).then(result => {
              resolve(result);
            })
            .catch(error => {
              reject(err);
              return;
            });
        } else {
          resolve(result);
        }

      })
    );

  }

  function linkUser(sub, event) {
    console.log("Linking user accounts with target sub: " + sub + "and event: ", event);

    //By default, assume the existing account is a Cognito username/password
    var destinationProvider = "Cognito";
    var destinationSub = sub;
    //If the existing user is in fact an external user (Xero etc), override the the provider
    if (sub.includes("_")) {
      destinationProvider = sub.split("_")[0];
      destinationSub = sub.split("_")[1];
    }
    var params = {
      DestinationUser: {
        ProviderAttributeValue: destinationSub,
        ProviderName: destinationProvider
      },
      SourceUser: {
        ProviderAttributeName: 'Cognito_Subject',
        ProviderAttributeValue: event.userName.split("_")[1],
        ProviderName: event.userName.split("_")[0]
      },
      UserPoolId: event.userPoolId
    };
    console.log("Parameters for adminLinkProviderForUser: ", params);
    return new Promise((resolve, reject) =>
      cognito.adminLinkProviderForUser(params, (err, result) => {
        if (err) {
          console.log("Error encountered whilst linking users: ", err);
          reject(err);
          return;
        }
        console.log("Successfully linked users.");
        resolve(result);
      })
    );
  }

  console.log(JSON.stringify(event));

  if (event.triggerSource == "PreSignUp_SignUp" || event.triggerSource == "PreSignUp_AdminCreateUser") {

    checkForExistingUsers(event, false).then(result => {
        if (result != null && result.Users != null && result.Users[0] != null) {
          console.log("Found at least one existing account with that email address: ", result);
          console.log("Rejecting sign-up");
          //prevent sign-up
          callback("An external provider account alreadys exists for that email address", null);
        } else {
          //proceed with sign-up
          callback(null, event);
        }
      })
      .catch(error => {
        console.log("Error checking for existing users: ", error);
        //proceed with sign-up
        callback(null, event);
      });

  }

  if (event.triggerSource == "PreSignUp_ExternalProvider") {

    checkForExistingUsers(event, true).then(result => {
        console.log("Completed looking up users and linking them: ", result);
        callback(null, event);
      })
      .catch(error => {
        console.log("Error checking for existing users: ", error);
        //proceed with sign-up
        callback(null, event);
      });

  }

};

I have been fiddling around with the same issue for a bit. Accepted answer sort of works but does not cover all scenarios. The main one is that once the user signs up with the external login, they will never be able to sign up with a username and password. Currently, Cognito does not allow linking Cognito users to external users.

My scenarios are as follows:

Scenarios

1. When the user signs up with a username password and signs up with an external provider, link them.
2. When the user signs up with an external provider allow them to signup with a username and password.
3. Have a common username between all linked users to use it as a unique id in other services.

My proposed solution is to always create the Cognito user first and link all external users to it.

Proposed solution

1. user signs up with username/password first then with an external user. No dramas, just link the external user with the Cognito user.
2. user signs up with external user first then wants to sign up with username/password. In this scenario, create a Cognito user first then link the external user to this new Cognito user. If the user tries to signup with a username/password in the future, they will get a user already exists error. In this case, they can use the forgot password flow to recover then log in.

const {
  CognitoIdentityServiceProvider
} = require('aws-sdk');


const handler = async event => {
  const userPoolId = event.userPoolId;
  const trigger = event.triggerSource;
  const email = event.request.userAttributes.email;
  const givenName = event.request.userAttributes.given_name;
  const familyName = event.request.userAttributes.family_name;
  const emailVerified = event.request.userAttributes.email_verified;
  const identity = event.userName;
  const client = new CognitoIdentityServiceProvider();

  if (trigger === 'PreSignUp_ExternalProvider') {

    await client.listUsers({
        UserPoolId: userPoolId,
        AttributesToGet: ['email', 'family_name', 'given_name'],
        Filter: `email = "${email}"`
      })
      .promise()
      .then(({
        Users
      }) => Users.sort((a, b) => (a.UserCreateDate > b.UserCreateDate ? 1 : -1)))
      .then(users => users.length > 0 ? users[0] : null)
      .then(async user => {
        // user with username password already exists, do nothing
        if (user) {
          return user;
        }

        // user with username password does not exists, create one
        const newUser = await client.adminCreateUser({
            UserPoolId: userPoolId,
            Username: email,
            MessageAction: 'SUPPRESS', // dont send email to user
            UserAttributes: [{
                Name: 'given_name',
                Value: givenName
              },
              {
                Name: 'family_name',
                Value: familyName
              },
              {
                Name: 'email',
                Value: email
              },
              {
                Name: 'email_verified',
                Value: emailVerified
              }
            ]
          })
          .promise();
          // gotta set the password, else user wont be able to reset it
          await client.adminSetUserPassword({
              UserPoolId: userPoolId,
              Username: newUser.Username,                                                      
              Password: '<generate random password>',                                                       
              Permanent: true
          }).promise();
    
          return newUser.Username;
      }).then(username => {
        // link external user to cognito user
        const split = identity.split('_');
        const providerValue = split.length > 1 ? split[1] : null;
        const provider = ['Google', 'Facebook'].find(
          val => split[0].toUpperCase() === val.toUpperCase()
        );

        if (!provider || !providerValue) {
          return Promise.reject(new Error('Invalid external user'));
        }

        return client.adminLinkProviderForUser({
            UserPoolId: userPoolId,
            DestinationUser: {
              ProviderName: 'Cognito',
              ProviderAttributeValue: username
            },
            SourceUser: {
              ProviderName: provider,
              ProviderAttributeName: 'Cognito_Subject',
              ProviderAttributeValue: providerValue
            }
          })
          .promise()
      });
  }

  return event;
};

module.exports = {
  handler
};

If you want to allow the user to continue login with email & password ("Option 1: User Signs Up with Username and Signs In with Username or Alias)") besides identity provider (google, facebook, etc) then the accepted solution won't be enough as Cognito can only have one email as verified.

I solve this by adding a Post Confirmation trigger which automatically verify user email if needed:

const AWS = require('aws-sdk');
const cognitoIdp = new AWS.CognitoIdentityServiceProvider();

const markUserEmailAsVerified = async (username, userPoolId) => {
  console.log('marking email as verified for user with username: ' + username);
  const params = {
    UserAttributes: [
      {
        Name: 'email_verified',
        Value: 'true'
      }
      // other user attributes like phone_number or email themselves, etc
    ],
    UserPoolId: userPoolId,
    Username: username
  };

  const result = await new Promise((resolve, reject) => {
    cognitoIdp.adminUpdateUserAttributes(params, (err, data) => {
      if (err) {
        console.log(
          'Failed to mark user email as verified with error:\n' +
            err +
            '\n. Manual action is required to mark user email as verified otherwise he/she cannot login with email & password'
        );
        reject(err);
        return;
      }
      resolve(data);
    });
  });

  return result;
};

exports.handler = async (event, context, callback) => {
  console.log('event data:\n' + JSON.stringify(event));

  const isEmailVerified = event.request.userAttributes.email_verified;
  if (isEmailVerified === 'false') {
    await markUserEmailAsVerified(event.userName, event.userPoolId);
  }

  return callback(null, event);
};

Note: This doesn't seem standard development or common requirement so take as it.

In aws-sdk-js-v3 I'm using @subash approach. I find that when you make an error callback, no extra user is created. Just the one that you create with your email.

const {
  CognitoIdentityProviderClient,
  ListUsersCommand,
  AdminCreateUserCommand,
  AdminLinkProviderForUserCommand,
  AdminSetUserPasswordCommand,
} = require('@aws-sdk/client-cognito-identity-provider')
const client = new CognitoIdentityProviderClient({
  region: process.env.REGION,
})
const crypto = require("crypto")

exports.handler = async(event, context, callback) => {

  try {

    const {
      triggerSource,
      userPoolId,
      userName,
      request: {
        userAttributes: { email, name }
      }
    } = event

    if (triggerSource === 'PreSignUp_ExternalProvider') {

      const listParam = {
        UserPoolId: userPoolId,
        Filter: `email = "${email}"`,
      }
      const listData = await client.send(new ListUsersCommand(listParam))


      let [providerName, providerUserId] = userName.split('_')
      providerName = providerName.charAt(0).toUpperCase() + providerName.slice(1)

      let linkParam = {
        SourceUser: {
          ProviderAttributeName: 'Cognito_Subject',
          ProviderAttributeValue: providerUserId,
          ProviderName: providerName,
        },
        UserPoolId: userPoolId,
      }

      if (listData && listData.Users.length > 0) {

        linkParam['DestinationUser'] = {
          ProviderAttributeValue: listData.Users[0].Username,
          ProviderName: 'Cognito',
        }

      }
      else {

        const createParam = {
          UserPoolId: userPoolId,
          Username: email,
          MessageAction: 'SUPPRESS',
          UserAttributes: [{
            //optional name attribute. 
            Name: 'name', 
            Value: name,
          }, {
            Name: 'email',
            Value: email,
          }, {
            Name: 'email_verified',
            Value: 'true',
          }],
        }
        const createData = await client.send(new AdminCreateUserCommand(createParam))

        const pwParam = {
          UserPoolId: userPoolId,
          Username: createData.User.Username,
          Password: crypto.randomBytes(40).toString('hex'),
          Permanent: true,
        }
        await client.send(new AdminSetUserPasswordCommand(pwParam))

        linkParam['DestinationUser'] = {
          ProviderAttributeValue: createData.User.Username,
          ProviderName: 'Cognito',
        }

      }
      await client.send(new AdminLinkProviderForUserCommand(linkParam))
      //throw error to prevent additional user creation
      callback(Error('Social account was set, retry to sign in.'), null)
    }
    else {
      callback(null, event)
    }
  }
  catch (err) {
    console.error(err)
  }
}

However, it is a bad UX as the first sign in with federated identity will only create the user but not allowing it to authenticate. However, the subsequent sign in with federated identity will show no such issue. Let me know, if you get any other solution for that first sign in.

It's also useful to keep email_verified as true so that user can recover their password. Especially true if you are using aws-amplify authenticator. This should be in your post authentication trigger.

const {
  CognitoIdentityProviderClient,
  AdminUpdateUserAttributesCommand,
} = require('@aws-sdk/client-cognito-identity-provider')
const client = new CognitoIdentityProviderClient({
  region: process.env.REGION,
})

exports.handler = async(event, context, callback) => {

  try {

    const {
      userPoolId,
      userName,
      request: {
        userAttributes: { email_verified }
      }
    } = event

    if (!email_verified) {

      const param = {
        UserPoolId: userPoolId,
        Username: userName,
        UserAttributes: [{
          Name: 'email_verified',
          Value: 'true',
        }],
      }
      await client.send(new AdminUpdateUserAttributesCommand(param))

    }
    callback(null, event)
  }
  catch (err) {
    console.error(err)
  }
}