As best I can tell, there are no authentication services that support unique usernames spanning local and social users! Certainly none with a free tier.
If you want to build an app with GitHub-style URLs and support social logins, you need to build the authentication yourself...
There are certainly ways to fake it - e.g. storing a separate property which is the "URL Slug" - but not as the local user's username. And trying to use usernames alongside that fails when someone new signs up with a username that collides with someone else's existing URL Slug...
Certainly, this doesn't seem to be possible with @auth0, @Firebase or #cognito. There may be others out there that can do this - in which case, someone please point them out to me!