To use google authenticator you do not need to give out your number. Most sites will have a qr code you can scan with your phone and the google authenticator app uses that to generate 2fa codes that are valid within a certain time frame.
You cannot skip this step on GMail, or at least I couldn't find how. I know you can use the app afterwards, but not before.
Other sites just give me the code, as you say. But not GMail.