If you get some Dry Erase marker on a mirror you can remove the dry erase marker using some acetone nail polish remover or some lacquer paint thinner.
Even some lighting fluid or rubbing alcohol should easily remove the dry erase marker off the mirror.
I would not recommend writing on the mirror or any other glass with dry erase marker because the dry erase marker can stain the mirror or glass.
But you can get the dry erase marker off the mirror or glass using the nail polish remover, lighting fluid, rubbing alcohol or even the paint thinner.
I've used nail polish remover to remove dry erase marker from my mirror before and it worked just great.