Naturally this issue is a loophole, 365 is asking the user for an Auth key from the app but the app is also asking for an auth key.
Depending on what make phone the user has this resolution can differ - essentially what you want to do is fully close down the application.
Apple: Double press the 'home' button - this will allow you to swipe left/right to see what apps are open currently. Swiping an app from the bottom upwards off the top off the screen will close it fully.
Samsung: At the bottom of the device screen there is a icon that shows 3 vertical lines together. Tap this and this will allow you to swipe left/right to see what apps are open currently. Swiping an app from the bottom upwards off the top off the screen will close it fully.
Re-opening the app following this should then no longer prompt you to enter a auth key to see the Microsoft Authenticator app interface.