Microsoft announced account backup and recovery today for its Authenticator app on iOS. In the app's settings, there's a toggle for Auto Backup, which will sync all of your accounts. When you sign into the app on a new device, you'll automatically see all of the accounts that you had set up.
The data that's being backed up is encrypted using your Microsoft account, but then the backup is stored in iCloud. This is an interesting method, as it doesn't use Microsoft's own cloud services like OneDrive. This also seemingly presents a hurdle for users that are going to want to use this feature to switch between iOS and Android devices, or vice versa. Microsoft says that the reason for this was that developing for iCloud simplified development.
The version number of the app that has the new feature is 5.7.0, and it's only available for beta testers. You need to be invited via TestFlight to try out the beta, but you can apply for an invitation here. The feature will be available for Android devices later on this year, but the company didn't clarify if it will be using Google Drive or something else for that.