Authenticator backup should allow cloud backup to work or school accounts and allow recovery across devices.
Presently, I cannot recover my work codes, via my personal account on an iOS device because the backup was originally taken on my Android phone.
A backup that can't be restored is useless (if it's even that good).
Khader Welaye commented
I second this, if I could backup using my Azure AD account then I would be putting all of my 2FA credentials in Microsoft Authenticator instead of LastPass