Thank you for reporting this issue.
I ping @Aurelsicoko and @alexandrebodin about this issue.

There is a workaround for this issue.

DO NOT update the extensions/users-permissions/models/User.settings.json to update the require: false of the username field.
This field could be used for the auth.

BUT you can create a new field name for example.
And the use the customization concept to update this function as you did in your Code snippets section.

I agree, as I tested the registration I found users when they register they could easily find their username was already taken. An option to make this non-unique or have another field as the username would be nice to have.

We already discussed this in another issue (can't find it) I think we should use the email as unique identifier and let the username be whatever the user wants.

I think that email could be the unique identifier. It would even be interesting if, when registering a user via Facebook (or other provider), he checked if the same email address already exists in the registered database to associate Facebook Login with the already registered user.

Is anyone up to this issue? If not, I can work on this issue if that's okay?

Is anyone up to this issue? If not, I can work on this issue if that's okay?

Do a PR =]

I added a PR for this. In short itis an added check in Providers.js that checks if the username is taken. If it is taken it adds a random suffix to the username.
I opted for tilde (~) as the separator as that works best with "real" usernames. Facebook names would look better with a psce and a hash(#) but facebook is not using the "real" username atm, I think it would be better to use the username (the one in the title bar on a profile page) for username from Facebook, but there might be a problem with this?

Anyway if anyone ends up here and want the quick fix:
Create a file: extensions/users-permissions/services/Providers.js
Add this to the top:

const crypto = require('crypto');

// Same random suffix as used in upload
const randomSuffix = () => crypto.randomBytes(3).toString('hex');

And this between "Retrieve default role" and "Create the new user":

        // Check if a user with the username exists
        var usernames = await strapi.query('user', 'users-permissions').find({
            username: profile.username,
          });

        // Add a new #randomsuffix to the username till we find one that is not taken
        var newUsername;
        while(!_.isEmpty(usernames)) {
            newUsername = `${profile.username}~${randomSuffix()}`;
            usernames = await strapi.query('user', 'users-permissions').find({
                username: newUsername,
              });
        }

        // Set the updated username
        profile.username = newUsername;

@derrickmehaffy I would like to work on this issue.