Several months ago, Google outlined a change that was going to be happening for Google Play Games. That change mostly circled around the removal of the Google+ account requirement for sign-in and the establishment of Gamer IDs, ala Microsoft Xbox. The update to the Play Games app for Android that supports this change is now rolling out in the Play Store, giving users the opportunity to setup their Gamer ID and take advantage of the automatic sign-in process for games.
The updated version of Google Play Games you are looking for is 3.7 and when you get the update to your device, you will have the option to create a Gamer ID by tapping on your profile avatar
and tapping the edit icon. You will also find that there are new permissions to setup that will enable the auto sign-in process for games going forward. You can opt out of this if you choose. Also, you can setup your account to be publicly searchable by your name or email address. Again, this is something that you control so if you don’t want that to happen, you can prevent it. Once you have setup your Gamer ID and setup your permissions, you are pretty much done.
Because Google Play Games comes on devices by default, getting this update is a matter of the OTA update coming to you. The update is in the Google Play Store so it should be coming to you over the course of the next few days.