Currently, the Get Videos endpoint does not return the game_ids for any of a streamer’s videos. This was available in the Kraken version of the API and has been pointed out as missing in Helix several times  . I checked the developer roadmap  and didn’t see anything promising related to this issue.
Is the API dev team working on adding this parameter back? When this question has been asked before, Twitch staff has said they were working on it.
: November 17, Helix API changes from Kraken - data missing
: March 18, Videos and game_id