For the same reason as we wake up, speak up, line up, dress up, think up ideas, brighten the place up or light up a cigar...or someone puts himself up for election or it's up to him to do something and so forth.
To quote..."If you are up to it, you might try building up a list of the many ways in which up is used. It may take up a lot of your time, but if you don't give up you may wind up with a thousand."
The answer is simply that it is an idiom...ie just the way we say/write things.
But slow up and slow down mean exactly the same thing, both examples having been in use since the 19th century! It's pretty clear that there really is no 'directional' significance in the word 'up' in such circumstances.
Actually if you consider the source of the water in washing up as opposed to washing down you will see that one does the up action when the source is below the objects being washed.
Chopping down from a height. Chopping up into a pile of wood.
Drinking down and drinking up don't quite comply.
Any guesses as to the part of speech when we use up or down in this way?
Well, personally, I do the washing-up with the dishes in the water; that is, when both dishes and water are on the same level. I lift a plate up out of the bowl only after it's clean. I still claim there is no directional significance involved here whatsoever.
Up may be a noun, adjective, verb, adverb and preposition. Given that all the examples offered above put it in direct relationship with a verb, it is essentially an adverb.