Nope, as long as you have registered or scanned your 2FA from Kraken to the Google Auth or any Authenticator you are using, that's how it's going to be. But with the looks of what you've said, it seems that you haven't connected it yet, maybe it's not a 2FA through google auth but through email to receive your codes?

Yeah, that's a security measure that most of the exchanges has but you can contact them and try to prove that it is you the owner of that account. But with your situation, they seem taking forever to help you out.