Google today announced an update to Google Authenticator -

This is the first update in a very long time, and adds OTP account sync to your logged-in Google account (with the ability to still use it without an account if desired).

The lack of account backup is what's kept me on Microsoft Authenticator for so long; I'm terrible for swapping phones frequently and adding the 10s of accounts back in manually to authenticator is nightmare material.

I know what I'll be doing now this week :)

Grab Google Authenticator via Google Play. The update is still rolling out so may not be visible just yet -