Game Center and iMessage for iOS9 vs. iOS10 multiplayer game?
Well, after a bunch of back and forth with the Quatrain folks, I finally heard this from them:


  After a lot of investigation, it looks like there is no simple way to
  add friends between iOS 9 and iOS 10 devices. We will be working to
  integrate the new iOS 10 Game Center functionality, but it won't be
  ready for a while as Apple made some drastic changes to the Game
  Center functionality.
  
  As a workaround, you can play the Auto-match games and if you select
  the Auto-Match Length of 5, you are almost guaranteed to only play
  each-other as most players don't use that option.


My friend and I tried the simultaneous "Auto-Match Length of 5" option and were able to play backgammon together. While playing there are "send friend" invite options but mine "went out" to who knows where and theirs (iOS10) at least gave a message saying not available.

So, sounds like there's some work to do for app developers to maintain the new style of Apple's Game Center functionality with non-iOS10 devices.