Restrict Game Center

To restrict Game Center features:

1. Go to Settings and tap Screen Time.
2. Tap Content & Privacy Restrictions, then tap Content Restrictions.
3. Scroll down to Game Center, then choose your settings.

You can restrict these Game Center features:

• Multiplayer Games: Prevent the ability to play multiplayer games
• Adding Friends: Prevent the ability to add friends to Game Center
• Screen Recording: Prevent the ability to capture the screen and sound

Taken from Apple's Support Page.