I uninstalled the Google Play Games app on my phone to get rid of the other annoying prompt, but now I'm left with this one. Is it possible to remove the prompt shown above?
It's really frustrating to see that pop-up every time we play a game. There's a solution if You've a rooted phone with Xposed installed in it. It's a Xposed module which will stop that pop-up. Be careful if not rooted.
No, some games as well as some apps need the Google Play Games and Google Play Services' latest version. So it may ask you to update or use to authenticate.
The entire data of the game is stored in your mobile and Google server. The game maker's server didn't store any score or data. So it needs you to install Play Services not only for this but also to use some API calls to your account info.
You cannot remove it.
I went into Androud Notifucations menu and dusabled notifications from the Google Play Store and that did the trick for me.
Not all apps depend on Google Play installation, despite asking for it. If I press Cancel when the prompt pops up, the game still loads and I'm able to play. But I can't get rid of the pop-up. It disappeared after a long while of ignoring it on other apps, but this new game just keeps asking for it. It also loads and I'm able to play.
Tell the game developer to correct his sign in algorithm. If the game prompts for Play Games install, the developer set it to prompt for it. It is an extra command to give to let you see this prompt.