で also can be used as "by" or "with".
手でつかむ grab by hand
箸で食べる eat with chopsticks
So that ゲームであそんでいます means you are playing with some kind of game ex."video game".
ゲームをあそんでいます is understandable however it doesn't sound natural.
You can say ゲームをしています is more natural.
If I hear ゲームをあそんでいます that gives me a little sense of toying with ゲーム.
The reason why we don't say ゲームをあそぶ maybe the word ゲーム connotes "play the game" in Japanese so we simply say "do (play) the game" ゲームをする.
you would like to put emphasis on あそぶ, you say ゲームであそんでいます
you would like to put emphasis on ゲーム, you say ゲームをしています