Below are the steps to setup your Authenticator accounts to backup to your iCloud.
First create a personal Microsoft account if you do not already have one.
Go to Microsoft.com
Click the avatar in the upper right corner
In the sign in box click Create One
Click get a new address
Complete the questions: Enter a username, Create a password, Enter your Full name
On your Iphone
Make sure iCloud Drive is enabled
Under iCLoud “Apps using iCLoud”, make sure Authenticator is enabled
Open Authenticator
Select the 3 bars in the upper left corner, then Settings
Under Backup, turn on iCloud Backup
You may receive a message that says “You need a personal Account”
Here click on Add Account
Enter the details from the account you created on the first step
If prompted enter a recover email account and verify
The below Microsoft article describes how to restore the accounts if needed.