No. If the title of the game you have purchased does not have a “Xbox One & Xbox Series X|S” label or is not in the Xbox Series X|S category, you can only use this game on your Xbox One console at once. If the name of the game includes Xbox One, you can install it on the Xbox One console, if the name of the game is called as Xbox Series X|S, you can install it on the Xbox Series X|S console.
