Google Authenticator after an iphone motherboard replacement
Google Authenticator stores your accounts in the iOS Keychain. There's nothing special about the way this is done - it is the same as with other apps on the iPhone.
When you restore an unencrypted backup, these types of Keychain entries are not included in the backup - and thus cannot be restored. It does not matter if it is restored to the same &quot;motherboard&quot;.
If you instead make an encrypted backup (including iCloud backup) and restore from that, your Keychain entries will be restored - and thus your Google Authenticator accounts.
This is entirely intentional from Google, and they have set up the app exactly to do this. If it weren't setup that way, your accounts could be compromised without having the &quot;second factor protection&quot;.
The mechanism through which the backup is “linked” to the actual device is entirely implemented by Apple - there’s nothing Google Authenticator-specific about it. The data is encrypted using the Secure Enclave’s unique is as part of the encryption key. The Secure Enclave is a special component of the processor on the iPhone that allows, amongst other things, encryption/decryption of data based on a unique ID that cannot leave the Secure Enclave.
This means that it is not possible to just read out the unique id to perform the decryption somewhere else. In practice this means that the data can only be restored on that physical device (I.e. the device having the same Secure Enclave as when the backup was made).
I know it is too late for you now, but when setting up 2FA accounts, you really want to set up at least 2 devices, so that you have a backup device in case one is lost for any reason.
If the services you use do not support having multiple 2FA codes, I would suggest instead using a different Authenticator app that allow syncing the secret behind the 2FA code between devices. The free Authy app from Twilio is a very popular choice.