IllegalSSL7research,
This also may be due to built-in multiplayer functionality of Game Center. If you're seeing other users at your school and it has an open network that is shared, other players near you would see your avatar and nickname and vice versa. To specifically disable that via Settings > Game Center, then tapping to turn off Nearby Players.
See more info from the Game Center Legal page here: Legal - Data & Privacy - Apple
"When you play a game that supports multiplayer functionality, other Game Center users who are accessing the same game on the same Wi-Fi network or local area network, or are within range of Bluetooth will be able to see that you are nearby and see your nickname and avatar. You can disable this feature on your iOS or iPadOS device by going to Settings > Game Center, then tapping to turn off Nearby Players. On Mac, go to System Settings > Game Center, and click to turn off “Nearby players.” On tvOS, go to Settings > Users and Accounts > [user name] > Game Center, then turn off Nearby Players."
Take care.