Running Apple Arcade Alongside Other Applications
This is a basic premise of computing.

Only one application can be frontmost at any given time. That is the one that will accept input from any connected devices, mouse, keyboard etc.

There is some leeway for such as the scroll-wheel, which can scroll any window the cursor is over, whether foremost or not, but that is an exception rather than a rule.
Also, such as a video or audio stream will continue to run, so you can watch YouTube or listen to the radio in the background

Games tend to alleviate the control issue by pausing when backgrounded - it saves your poor avatar from being squished/killed/juiced etc whilst you're not actually controlling it.

There is no way round this.