My situation is that I had two-factor authentication (2FA) set up on the Microsoft Authenticator app to log into my school email. Unfortunately, my phone was reset, all of my data was lost, and I hadn’t turned on the cloud backup feature. (I’m so frustrated that Microsoft Authenticator disables backup by default!)
Now, I cannot log in without the code, there are no other login options available, and I cannot ask an administrator for help. Is there any way to resolve this situation?
@Gainey_Kanchha — the two details in your post are what decide this: the backup was off, and you cannot ask an administrator. This is the whole answer, top to bottom.
About that backup first: on a school or work account, Microsoft Authenticator saves the account name only. The codes are not included in any backup. So the setting you are annoyed about would not have saved this account either — nothing was lost by missing it.
search your school's site → MFA reset
ask the help desk for → "Temporary Access Pass"
register the new phone → then sign in
1 · Search your school’s site for “MFA reset” before you call anyone. Several universities run a self-service reset that needs no administrator and no ticket — you prove it is you with a photo of your government ID. UNLV, NUS, Graz and JAMK publish theirs openly, so the pattern is common.
2 · No self-service reset on your school’s site? The phrase for the help desk is Temporary Access Pass. It is a time-limited passcode Microsoft built for exactly this — you sign in with it and register a new phone. Asking for “an MFA reset” sends a desk searching; these three words do not.
3 · Register your new phone, then sign in normally. Your old code is not needed at any point in that process.
One thing to expect: an account allows five authenticator registrations. Meet a message about that limit? Ask the help desk to remove the old phone first.
Once you are back in, add a second method so the next reset phone does not stop you.
Open this page before anything else: mysignins.microsoft.com/security-info — or just type aka.ms/mfasetup, same place. 
Sign in with the school account and read “Sign-in methods”:
The backup mystery, settled — on a school or work account, cloud backup in Authenticator is controlled by the organisation, not by you. Plenty of tenants switch it off. You did not miss a setting; the school never allowed it. Your reset was not your mistake.
Say these two words to the help desk: “Temporary Access Pass.” A time-limited code that lets you sign in once and register the new phone. Some tenants let you request it yourself — look for Add sign-in method → Temporary Access Pass on that same page. A desk answers faster when you use the product’s own term.
While you wait — if a personal Microsoft account is linked, the account recovery form still applies; and the 25-character recovery code from setup works if you kept it. 
The day you are back in: register two methods (phone + passkey), and set self-service password reset so a wiped phone never locks you out twice.
!