A week and a half ago, Microsoft updated its Azure Authenticator app on iOS and Android to become Microsoft Authenticator. While the idea was to merge two factor authentication from Microsoft and Azure AD accounts into one app, the iOS version wasn't quite complete.
If you're on the standard version of the app - which is version 4.0.3 - you might attempt to add your Microsoft account and find that the app asks you to scan a QR code. With version 4.0.4, you'll instead see the screen that you see above on the left, allowing you to choose between a personal, work or school, and other account.
Another feature that's in today's beta build is push notifications. Here are the release notes:
This build includes session approval (push notification) for Microsoft accounts, as well as updates to the Microsoft Account (MSA) SDK.
If you already use your Microsoft Account within this app via OTP codes, you must re-add your account in order to take advantage of session approval. The recommended path is to make sure you have an alternate proof (sms or email) on your account, then remove the old account (swipe right from the accounts list) from Authenticator, and last, add a new Personal account.
If you'd like to be on the Microsoft Authenticator beta for iOS, you should still be able to do so. Just email firstname.lastname@example.org with your name and Apple ID and they'll set you up.
If you're not into betas, that's OK too. While no timeframe is provided for when a beta will be available to the public, you shouldn't have to wait too long.