If the user has the same Game Center login on all of their devices, then you should not need the actual Game Center ID.
The Game Center login command uses whatever is stored on their device to log them in (or asks for login info).
Unless they use a different Game Center identity on their different devices, then you will always be connecting to the same Game Center Account.
But, having just looked at your Google post, the issue comes down to multiple players on the same device.
The problem appears to be with Game Center and not necessarily AGK. It looks like only one Game Center ID can be logged in on the same device at any one time. So multiple players on the same device cannot be logged into Game Center at the same time. At least this is what I see in my googling.
And, I suppose, in your game's case, you want to be able to keep whatever game state constant across devices.
One way might be to ask for a user name when the game is installed (or, if you allow multiple users on one device, also allow adding a new user name). Then, since you appear to be using an internet account to store stuff (database, I expect), you do some back and forth to add/check the name and return a unique id for your system. This is what I am doing in my WIP. It even recognizes when someone uses the same name on multiple devices and verifies that it is them.
Cheers,
Ancient Lady
AGK Community Tester and AppGameKit Master