You would expect that if you restore the full backup of your iOS device to a new one, because you lost it for instance, that on the new device you could open the Authenticator app and see the same keys as you had on your old device. That is not the case though.

Under the hood Google Authenticator uses keys to generate the codes you see on screen and these keys are not backed up.

It’s a difficult decision of course. If you back them up in iCloud Apple and people who hack your Apple account have access. If you don’t the keys are lost if the device breaks or is lost and you need a workaround.

I think they do backup if you do an offline (e.g. iTunes) full backup and restore on the same device, which of course is never what people do.

No, I know because that’s what I do and I had this fail on me when I migrated to a new phone. I switched to a different app that does backup keys when you use an encrypted iTunes backup.

Edit: oh you wanted to restore on the same device. Well that might work but it doesn’t help when migrating or if your phone is not available.