How to Back Up your Microsoft Authenticator App
To be able to transfer all of the accounts in your Microsoft Authenticator app from one device to another, like when you get a new or different device, you will need to back up your accounts in Microsoft Authenticator.
First, you will need a personal Microsoft account to act as your recovery account and have either iCloud or Cloud backup turned on for the actual storage location on your mobile device. If you don’t already have a personal Microsoft account, you can get one for free here: Office 365 Login | Microsoft Office and click on Sign up for the free version of Office.
To back up your accounts, in the Microsoft Authenticator app find and click on Settings (under the hamburger menu for iOS or the three-dot menu for Android). Next under Backup, turn on either iCloud backup or Cloud backup depending on your device type. Instructions for backing up your Authenticator app accounts can be found by selecting the Learn more link under this same section on your phone.
When you get a new phone, transfer all the apps and data from your old device to your new one using whatever method you prefer. If the Microsoft Authenticator app did not install, make sure to install it. Then open and click the link that says something like Restore from backup.