The system works through a series of redirects. Say if I have to prove my identity to some government website, it offers different authentication methods, one of which is my bank. I 2F authenticate with my bank the way we agreed, then it redirects me back to the originating site.

Different banks offer different systems for 2FA. Most also offer an app. Google services are not always required. If you don't want to use an app, there are ways; in addition to username/password, usually a SMS for strong auth, or some code from a database that I have for weak auth.

No app, OS-independent.

If my bank forced my to use an app that only works with google services, I'd change banks. As I did, actually.