Re: "I don’t remember who is who in the Game Center friends.
Is there a way to know who is it I am playing with?"
Am no expert on using Game Center, yet your question deserves some response.
Seems your Game Center friends have the control of how much information they want to share.
Apple Support's article for iOS18 may be helpful:
Use Game Center to play with friends on iPhone - Apple Support
Eg: says:
"With Game Center, you can send friend requests, manage your user profile, earn achievements, compete on leaderboards, and play with friends during a FaceTime call. You can use Game Center to access supported games across all your Apple devices."
You can check if you have restrictions set: Settings > Screen Time > Content & Privacy Restrictions >
and
Apple provides documentation about the privacy practices for Game Centre:
This article: Legal - Game Center & Privacy- Apple
Eg: says:
"When your Game Center nickname is linked with your contact card, Game Center will use the email address and phone number associated with your Apple Account. This allows your Game Center friends to find you more easily based on the name they have for you in their Contacts app. You can control this setting on your iOS, iPadOS or visionOS device by going to Settings > Game Center > Help Friends Find You, and on Mac by going to System Settings > Game Center > Help Friends Find You."
There is a much more information in the linked articles.
