In the module you are developing, add the conflicting packages to peerDependencies (and remove them from dependencies or devDependencies):

  // package.json
  "peerDependencies": {
    "react": "16.13.1",
    "react-dom": "16.13.1"
  },

Execute npm install in your module.

Now add them to the webpack configuration of your module as externals. These packages shouldnt be included in the bundle of the module (the app that uses the module will provide them):

// webpack.config.js
module.exports = {
    /*
    rest of config...
    */
    output: {
        filename: "index.js",
        pathinfo: false,
        libraryTarget: 'umd', // In my case, I use libraryTarget as 'umd'. Not sure if relevant
    },
    externals: {
        // Use external version of React
        "react": {
            "commonjs": "react",
            "commonjs2": "react",
            "amd": "react",
            "root": "React"
        },
        "react-dom": {
            "commonjs": "react-dom",
            "commonjs2": "react-dom",
            "amd": "react-dom",
            "root": "ReactDOM"
        }
    },
};

Then, after building your module, in your application you can check that both versions are now the same:

// node_modules/mymodule/src/index.js
export { default as ReactFromModule } from 'react'

// src/index.js
import React from 'react'
import { ReactFromModule } from 'mymodule'
console.log(React === ReactFromModule) // true :)

In response to another comment, merely moving React to peerDependencies does not adequately resolve the issue in all cases. I would reply to that comment directly, but StackOverflow requires more reputation to respond to wrong answers than it does to post them.

I have a shared React component module built using Webpack and have run into the same issue. I've outlined one possible fix in this comment below which requires modifying peerDependencies and using npm link in a fashion similar to the answer shared by mtkopone. https://github.com/facebook/react/issues/13991#issuecomment-841509933

My solution is a bit hacky and I wouldn't recommend it for long-term use. If you are using Webpack (which you tagged this question as), this article may detail a more permanent solution (https://medium.com/codex/duplicate-copy-of-react-errors-when-using-npm-link-e5011de0995d). I haven't tried it yet, but the author seems to have tried all the (incorrect) solutions out there and is also running into the hooks issue while trying to build shared component libraries.

The author of that article is trying to debug a Create-React-App app. While CRA uses webpack under the hood, you can't access the webpack.config directly, so the author has to perform some workarounds to do so. If you aren't using CRA, but just plain Webpack, then you could consider using the resolve.alias section of webpack.config to ensure there are no duplicate copies of React (see: https://blog.maximeheckel.com/posts/duplicate-dependencies-npm-link/)

Adding react and react-dom as peerDependencies in the package.json didn't work for me.

I had to add an alias to the webpack configuration file:

// webpack.config.js
resolve: {
  alias: {
    react: path.resolve('./node_modules/react'),
}

I was attempting to use the peerDependencies and removal of the devDependencies and it was failing.

It turned out I had a node_modules folder in one of the parent folders of the library I was working on and the duplicate version of React was being loaded from there instead of the tool that was trying to use the React library.

Rather than editing the devDependencies to remove react I just wrote a small script to delete anything that's in the peerDependencies from the node_modules folder.

npm view --json=true . peerDependencies | jq -r 'keys | .[] |  @text' | while read dep; do  rm -r ./node_modules/${dep} && echo Removed ${dep}; done