I thought Google authenticator is encouraging online backup. That has been the reason I do not like it.
I don't think there's an online backup it does have an export option but you can only export them if you are going to move the authenticator to a new device because it would generate a QR code that you need to scan to another device. I never tried exporting my authenticator with email I saw this as another option but you can just ignore it and use the QR code option instead if you want to export and move it to a new device.
Actually their is an online backup where you will be required to permit the authenticator app to synchronize with any Gmail account linked to you device most times it uses your primary email by default to conduct the backup.