It "provides protection" by not backing any data up anywhere, so if the phone it is on is wiped, the codes can't be recovered.

It's a blessing and a curse, but with a bit of work (like backing up all codes at creation time) it's a non-issue for me that I actually appreciate.

Google Authenticator, the one time password (OTP) app made by Google, is storing data locally on the device. To my knowledge it has no link with your phone number.