I did not caulk our vanities (both furniture style) to the wall, for a variety of reasons that I will not confess, but mostly because I don't like the way it looks. If it's supposed to be furniture, then by golly, it shouldn't be caulked, IMHO.
However, the walls behind are tile and grouted completely, as is the floor, and I believe I have read that it may be code- some will caulk just for inspection ;-*
Q