It has overlapping plot in the same way Overwatch does. Technically the characters all have backgrounds, and that's what Arcane shows some of. The game may use that to give character to the heroes, but the game itself doesn't really add anything to it. The game is probably improved by the existence of the show (I assume, I never cared for it I'm general), but the show really doesn't depend on the game.

(Technically I think there are multiple games in this universe now. I can't tell you much more, but I think there's an action game with some pirate character at least. That may have more story, but it doesn't matter for Arcane.)