I didn't say you're completely wrong, either. :) (In fact, I stressed that both equations are of elliptic type, so solutions might look similar.)

One can make a similar heuristic argument, though, as to why the surface you observed should follow the minimal-surface equation.

EDIT, as you've updated your comment:

> Isn't Poisson's equation basically describing a minimal surface for small z?

Small changes in z, I think.