Going with one of the words you already propose, I would call it the dishwasher frame:
[Merriam-Webster]
4 a : a machine built upon or within a framework
// a spinning frame
b : an open case or structure made for admitting, enclosing, or supporting something
// a window frame
Note that a frame can be made of any material.
Housing might also work, but I'm used it it being used in a different context. (One involving habitation or smaller items.)
