Bathroom light switch
Some dimmers use the ground to trickle a small amount of current for normal operation. If yours is working without it connected, I would say it doesn't fall into this category (although you could provide the model name/number to be certain).

Most likely the ground connection is there simply because all residential switches, receptacles, etc. require a ground these days.

It sounds like a ground wire is available in your switch box, but it is going to your fan's switch. No, you cannot double-tap the ground screw on your other switch. You need to use some form of wire connector to make what is called a pigtail to connect the ground from each switch to your ground wire in your switch box.

Illustration of the pigtailing concept