Try knitting or maybe crocheting.
What you currently do is a manual thing. People usually do not have this as a hobby, since it is considered work. Actual work. You either get paid by the bank, supermarket or whatever your employer is. Or you might get a discount on the fee to transfer coin money into your account, if you would wrap it yourself. Such things.
Playing a game might not give you the satisfaction of keeping your hands occupied. There is no haptic, you need a concious effort, mostly visual. You might hear an audiobook or radio or hear the tv, while playing a game and there is even the idle genre.
But playing so while occupied otherwise is a bit different in my opinion. Games are actually made to keep both your attention and keep your hands busy. Especially any type of action game.
For games in general, if there is no new story, you can use any game to hear things while playing the game. It goes with knowing the game, that you can run it on low burn in your brain. Similar how most people can walk and talk at the same time.
If work or pseudo work in a game is relaxing for you, maybe you would like one of the many simulator games. https://itch.io/games/tag-working-simulator (I am not sure if the games I talk about are in this tag. Maybe they are not indie enough to be on Itch at all. There is a big market for these. I am talking about all those Truck and Farm and whatnot Simulator games. Amercian Truck Simulator has about 100000 Reviews on Steam.). If it is only about the lack of required focus, that might just be another word for casual games. https://itch.io/games/tag-casual
The thing is, video games usually are not made to be played while watching a show, but be played instead of watching a show.
You might like this very old game https://duckduckgo.com/?q=Donald+Duck%27s+Playground&ia=web (Watch a video of it. There is even coins in it, you need to handle.)