While you can connect your games to Google Play or Game Center it is not possible to use this method for signing in over more than one device, or switching devices (if you get a new one).
This does however provide an extra level a retrievability if you lose access to your game. For example, if your phone breaks and you were playing as a guest but linked to Game Centre or Google Play, the unique username of your account may help us locate your data in our database. It is always best to ensure you have your unique Runaway ID number written down somewhere too though!