Game IDs

Need unique game IDs for the game currently running. These game IDs must be:

- treated the same as user IDs (since users should only be in one game at a time).
- Be used to filter the appropriate chatrooms when requesting chat messages (multiple chatrooms can exist in one game).

Start by tracing how user IDs are represented and how chat messages and chatrooms are requested over HTTP or WebSocket. Identify the current game context, then verify that game IDs are handled like user IDs and that message requests select only chatrooms belonging to that game.

Written by the indexing model from the issue text.