Google Authenticator has added account transfer at some point in the recent couple years - you get a QR code that you can scan on another device to transfer stuff. It's not automated backup, but it makes migration to new devices easier.
I'm pretty sure the QR codes actually encode your secrets so if you want to print them out you're good to go. Obviously won't include any new ones you add in in the meantime though.
This is only for Google-specific services that use 2FA (and it works with any authenticator apps, not just Google Authenticator, because this functionality is dependent on the service, not the app).
So if I try to do this for my Google account 2FA (with Google Authenticator, Microsoft Authenticator, or literally any other authenticator app), it works. But it won’t work for any of my other 2FA accounts, like MSFT, Discord, etc., no matter which authenticator app I use.
To your defense, Google Authenticator added the option for restoring from local backup fairly recently, but only for Android. And even then, no ability to sync it with multiple devices.
That doesn't make much sense to me. If the app has the state necessary to generate TOTP codes, why couldn't it transfer that state to a different device? How could it be dependent on the service?
I recently changed phones from Android to iOS and the export/import from Google Authenticator was fine for non-Google services.
Edit: I think there was a bug where it seemed to only export the tokens visible on the page, so I had to scroll and do it again for additional tokens. Maybe you are hitting that.
