Should it be mandatory to hire a mechanic? or someone else with any kind of FAA license? or could anyone wash one?
Any yahoo can wash a plane...even a pilot.
The only regulation of which I am aware concerning washes involves maintenance. A maintainer must clean the aircraft in order to determine servicability of a component. The regulation does not specify who must do the cleaning, however. I doubt a FSDO (in the US) would read that as "A&P must clean it."
There is no regulation on who can wash a plane. You'll probably find that quite a few planes (at least of those used for instruction) are in fact washed by student pilots.