If you choose to log in through Google then you need a Gmail account, when you press the Google log in button the app searches for any Gmail accounts that you’re logged in with already on that browser or device and then gives you the option of which one you want to use. Your account will then be registered to that Gmail address, if you log in using a different Gmail address after that then you will create a new account.
However if you press the Facebook login button the app will search for a Facebook account that is logged in on that device or browser. Once it’s found one it will ask if you want to use it and if you accept it will search your Facebook account for the email address that’s attached to it. Screen Time will then use that email address to create your account and register it to that email address.