Microsoft released the first major update for the beta versions of its Authenticator app on iOS and Android, bringing the version numbers to 5.0.0 and 6.0.2, respectively. The update adds the ability to sign into your Microsoft account directly from your phone.
When logging in, you'll be able to select "use an app instead", and you'll then receive a push notification on your phone or supported wearable device (such as an Apple Watch). Once you choose to interact with it, you can enter your PIN, or use your fingerprint. It also might ask you to complete a 'challenge', which would ask you to repeat a number that you see on the main screen.
In order to enable the new feature, you'll need to be sure that you have a password, PIN, or fingerprint set up on your device, and then you can choose "enable phone sign-in" from the dropdown menu. If you're adding a new personal Microsoft account, however, the feature will be enabled automatically.
Microsoft Authenticator was originally released for iOS and Android back in August, and although it was promised within a week, didn't arrive for Windows 10 Mobile until November. Sadly, there was never a beta program implemented for Windows, but it seems likely that the new phone sign-in feature will arrive for Microsoft's mobile platform eventually.