I'm waiting for the day in which I can sign up for 2FA without giving my phone number. At this point, I believe they are holding it on purpose, as my phone number is a much more reliable unique identifier than my username and/or cookies.
(Yes, you can use the Google authenticator, but no, you can't do it if you haven't given your phone number first)
Edit: by "they" I mean GMail - other sites work just fine.
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 could create a Google Voice account with a free number and link to that. Set it up so calls/sms/etc go nowhere but can be changed if you need to restore your Gmail account later with that number.
How recently have you tried to do so? I tried a couple of months ago to set up a Gmail account with a Google Voice number for verification, and it refused to let me with a message which I recall as being vaguely like "This is not an acceptable verification number".
(Yes, you can use the Google authenticator, but no, you can't do it if you haven't given your phone number first)
Edit: by "they" I mean GMail - other sites work just fine.