I've been going through Apple's (awful) documentation for both MusicKit, the API and MusicKit JS, but I haven't been able to find an endpoint or method to retrieve a user's display name, email or any other information that I can use to identify the user.
Is there even a way to retrieve this?
I've been having the same issues as well. As far as I can tell, there is no way to identify the user. I'm getting around this by just giving users temporary sessions that get destroyed when they clear their cookies or unauthorize from musickit.
On the other hand, you could ask the user to log in with some other auth provider like google, github etc, and then authorize their apple music after that. Its's not the best user experience to log in to two things in a row, but I guess we have to make do