Remember, if it was actually anonymous, you wouldn't need Facebook's help to implement it.
People implement FB login for a variety of reasons, but one is that doing proper account management is hard. If you want to implement a non-FB anon login, you basically have to implement a full account management solution yourself, assuming you want the anon account data to be persistent (that is, it's not truly anonymous in the sense that you sign in, do things, sign out, and it's as if you were never there). This basically fixes one of my final issues with FB login, that it's still not clear what data FB will give to the app. If anon login means FB doesn't give anything to the app aside from an opaque ID token, that's pretty cool.