I once sat through a truly bad movie years ago (Close My Eyes) just because my heartthrob Alan Rickman was in it, and I was surprised to hear him refer to the dishwasher as "the washing-up machine." I had heard British friends refer to "doing the washing up", but I thought that they'd meant an overall cleaning of the dining area after a meal. It hadn't occurred to me at until I saw the movie (and then did further investigation) that "doing the dishes" was an AmE phrase, and "washing up" was the BrE equivalent.
Yes. When it comes to cleaning utensils and implements used for preparing and eating food, it's all "dishes." And "do the dishes" is understood to mean cleaning them.