I’m confused on why does 2FA on this casino requires email while it usually done using authenticator app like google authenticator without the need of email verification.
It doesn't, the email is just a confirmation that the payment has been sent.
It works just the way you describe it "using authenticator app like google authenticator without the need of email verification."
So far I have used 2fa with Google authenticator and have never been asked to verify the email first, except that when activating it is usually asked to verify the email perhaps just to make sure that you are the real owner, what freebitcoin does is probably to keep all its users safe from the undesirables, so far I use the freebitcoin account on my personal laptop so they never ask me again to log in to the account using Google's 2fa authenticator and keep my account open there unless I log out of my account, maybe they will ask me to re-request the Google authenticator code to log into my account.
But it's still somewhat unclear what the OP means, perhaps whether he wants to withdraw money or activate the Google authenticator, usually when withdrawing it's just an email notification of the withdrawal process, not some kind of email confirmation, it's been a long time since I activated the Google authenticator, as far as I know, if they ask to You verify your email perhaps just to make sure that you are a genuine user, that's actually not a bad thing either.