userList[0] in the function User will return an array: ['name', 'surname']. To get the first name for example, you need to use this.name = userList[i][0]

function UserList() {
    let users = [];
    while (true) {
        let response = prompt('Please, enter your name surname?');
        if (response == null) {
            alert('cancel');
            break;
        }
        users.push(response.split(' '));
    }
    return users;
}

function User() {
    for (var i = 0; i < userList.length; ++i) {
        this.name = userList[i][0];
        this.surname = userList[i][1];
        this.regDate = new Date;
        console.log('Name: ' + this.name + ' Surname: ' + this.surname + '. Date of registration : ' + this.regDate)
    }    
}

let userList = new UserList();
let user = new User();

1. UserList shouldn't be a constructor. It should just be a function that returns an array of names.

2. You shouldn't be iterating over the list of users within User. You should then be iterating over the array creating one new User on each iteration which should be generated from a constructor. You can just pass in each name from the array and build an new object.

function getNames() {
  const users = [];
  while (true) {
    const response = prompt('Please, enter your first and last names');
    if (response == null) break;
    users.push(response.split(' '));
  }
  return users;
}

// Pass in a user name from the array as an argument
// It's array so we can destructure the first and last name
// immediately
function User([first, last]) {
  this.first = first;
  this.last = last;
  this.regDate = new Date();
  this.message = `Name: ${this.first}. Surname: ${this.last}. Date of registration: ${this.regDate}.`;
}

// Iterate over the array generated by `getUsers`
// and for each name create a new user.
for (let name of getNames()) {
  console.log(new User(name));
}

Additional documentation

• Destructuring assignment

You are pushing an array in an other array, so your index is not correct (array looks like this: [["firstname", "lastname"]]). You could spread the items when pushing using the spread operator (...), you could also flatten the array using flat().

Also when creating a date, use new Date().

 function UserList() {
   let users = [];
   while (true) {
     let response = prompt('Please, enter your name surname?');
     if (response == null) {
       alert('cancel');
       break;
     }
     users.push(...response.split(' ')); // flatten

   }
   return users;
 }

 function User() {
   this.name = userList[0];
   this.surname = userList[1];
   this.regDate = new Date(); // ()

   console.log('Name: ' + this.name + ' Surname: ' + 
     this.surname + '. Date of registration : ' + this.regDate)
 }

 let userList = new UserList();
 let user = new User();

Using flat()

return users.flat();

Edit

I actually understood the question wrong (thought you only wanted 1 user), the other answer should be correct and makes more sense.