Apple has removed the Game Center app from the first betas of iOS 10 and macOS Sierra. The change appears to be permanent, according to the iOS 10 Preview release notes. Although it's still available in Settings, “the Game Center app has been removed,” reads the document.
That doesn't mean that developers can no longer use the Game Center service or implement Game Center features—quite the contrary—it's just that the user-facing Game Center app is no longer available.