Is there some kind of backup for google authenticator?
~snip~
And of course, as I mentioned above, good 2FA apps will let you export an encrypted database of all your codes which can be used as a back up.
So I guess Google Authenticator must be a bad 2FA app as far as this standard is concerned.
I took a quick search if it is possible for a Google Authenticator user to export an encrypted database of the codes to be used as a back up. It seems it's not at all possible. What is allowed in Google Authenticator is to export 2FA accounts in another device. That will serve as a backup but you need to have another smart phone for that. As opposed to the one you mentioned, a USB stick is enough.
ETFbitcoin mentioned about obtaining the database file right from the device's storage itself, although it requires an app that reads SQLite files. I don't know if this is possible with iOS, though.