Microsoft account supports TOTP. You don't need to use Microsoft's authenticator app, you can use any that supports TOTP. Google Authenticator is unlikely to send your phone number to Microsoft.
Indeed, but some of my test accounts setup using TOTP still have been hit by the phone number prompt, where this poster didn't- so they might treat their own authenticator higher. (And if they do, i have to wonder if it means it sends more data back than others do)