After initiating a local test with 2 clients, the second client returns 0 for game.GameId. If you initialize a third client while the test is running, the game.GameId returns correctly. It is consistently reproducable under these instructions.
This is a headache for testing and it appears to have rolled out with the most recent studio update.
No beta features are enabled. This has been reproduced on Windows 10 and Windows 11.
Hey OP - closing this out. Thanks so much for the report. Would you mind posting again with a bit more info? Details on system, screenshots, anything necessary. See above.