They don't offer any backups (at least on iOS) and as a result, if you lose your phone, you are hosed. Google Authenticator also doesn't use iCloud for backup for files like other apps. I also just assume at this point no one owns that app and that it'll never get backups because that's how Google operates.

I've seen multiple people lose their TOTP codes this way and have been locked out of their accounts. Or even the more simple case, they buy a new phone, restore from backup and just assume everything is peachy then send their old phone back and then don't realize it until they open the app for the first time.

Use something with cloud backups for your safety.