Hi, I have a wood floor to ceiling cabinet in this house I bought.
Inside, underneath a shelf, is a 14 gauge Romex line that goes from
the wall on the left to a box with an outlet on the right. The Romex
is stapled to the underside of one of the shelves.

Is this even close to being code ?

Does the wire need to be run through a conduit ?

I've seen other wires just stapled to a board on a wall, like the 110
line for a washer in a basement.

Thank you.