Is GameID a stable Id?

Yes but also no.

Twitch uses data from IGDB to populate their own categories with.

There are some cases where a game name of ID foo will get deleted from Twitch for whatever reason.
Then get readded to Twitch with a new ID of bar.

Generally speaking gameID’s should be stable.
But I only tend to cache this data for 24 hours and refetch as needed.