I'm a bit concerned about the message this app is giving me. It is saying that my currently installed version is no longer supported. Why am I getting this message directly from the app, shouldn't the play store be telling me that an update is available to update this app?
For some reason Google have created a whole new app for the new version of the Authenticator app:
See also Why does the Google authentication app have “obsolete” in its title on Play Store? and in the news Android Police: PSA: Google's Authenticator Updated To v2, Except It's A Brand New App, And You Need To Install It To Get Future Updates (Old One Is Dead)
It is worth noting that as long as you still have the old version installed when you install and first open the new version, it will offer to import your old settings and accounts. If you remove the old version before running the new one you'll need to re-add your accounts, which may involve an annoying process of first turning off two step auth, and then re-enabling it on those accounts.