What a black hole looks like is independent of coordinates - what an observer "sees" or measures does not depend on what coordinate system they choose to make their measurements in.
However, many measurements (with notable exceptions like the spacetime interval) are not invariants for observers in different frames of reference. That is not a problem and there is no "true" reference frame.
Thus a distant observer "sees" (actually they don't because of redshift) material accumulate towards the event horizon and that is true whether they adopt Droste, Kruskal-Szekeres, Gullstrand-Painleve or any other coordinate system.
Ditto, a falling observer will fall through the event horizon in a finite proper time, no matter what coordinate system they choose to use.