Is there a difference between playing in Chromecast vs playing in Chrome browser? And why do you need a controller at all if you're just playing desktop games?
Playing on Chromecast is for playing on TV. Chrome is for playing on PC.
You need a controller presumably because you need a way to provide input if you're playing on TV. On PC you can use keyboard and mouse but Chromecast doesn't have a way to connect to keyboard and mouse as far as I know
IF you want to play on a windows pc/chromebook/... you can use most controllers, e.g. dualshock, xbox etc, or choose to not use a controller.