For those of you who are having issues with the iOS native mail app, that's a separate issue because of limitations set by the iOS platform. It's covered in our FAQ: aka.ms/AuthAppFAQ.
I've copied the pertinent information down below.
"While signing in to my work or school account using the default mail app that comes with iOS, I get prompted by Authenticator for my security verification information. After I enter that information and return to the mail app, I get an error. What can I do?
This most-likely happens because your sign-in and your mail app are occurring across two different apps, causing the initial background sign-in process to stop working and to fail. To try to fix this, we recommend you select the Safari icon on the bottom right side of the screen while signing in to your mail app. By moving to Safari, the whole sign-in process happens in a single app, allowing you to sign in to the app successfully."