Why do we qualify "dish washer" or "car wash" but not "clothes washer"?
Solution 1:
I suppose because it was the first one.
We, in the UK, had washing machines (for clothes) long before dishwashers and car-washers. So that by the time those machines came along, the term 'washing machine' had already been claimed.
It is also the case (at least in Britain) that washing clothes has always been known as 'washing'. A question like: 'When do you do your washing?' almost certainly refers to washing clothes, not to washing dishes etc (that's called 'washing-up') nor the washing of oneself, or the baby (that's bathing, or showering).