The same applies to the trim in a 737 and any other transport class aircraft. If you move the control column in the opposite direction of trim, the electric trim and autopilot trim will cut out. The same applies to the autopilot in all types I've flown, it disconnects if you override it manually (except for CWS mode, but that's known and expected)
I don't fly the 737, so I haven't read the aircraft specific books and systems overviews, but I expect this override behavior is very confusing in the MCAS scenario. Because any pilot would counteract a nose-down force by pulling on the yoke. The MCAS would then stop moving the trim, and the pilot would think (as would be true in any other situation) that the override worked. But MCAS didn't stop because of the override, it stops because it always does that after a few seconds and will re-activate 10 seconds later (or 20, didn't read a definitive number on this).
The confusing thing is that 95% of the unwanted trim movements are caused by the autopilot, so at this point the pilot would think the situation is under control and probably disconnect the autopilot if the movement didn't already,
The other 5% (or probably less) of unwanted trim movements are trim runaway, which is either a stuck switch (2 actually, you depress two switches at once to make it move, pressing 1 does nothing) or an electrical problem.
In both cases pulling back works and stops it. If it stays "stopped" you think autopilot and leave it disconnected. If it keeps moving continuously you think trim runaway and flip the trim cutout switches or pull the circuit breaker (this one is usually marked bright orange so you can find it quickly in planes without cutout switches)
But in the MCAS case, and without MCAS knowledge (which apparently nobody had) you would not expect it to start moving again later. You would have concluded "autopilot" of the above two scenarios, and after disconnect would think you're safe. So now when it starts moving again, your first idea is that the autopilot did not disconnect which leads to even more confusion in the cockpit.
The LionAir flight that was saved by the jumpseater got lucky in the sense that this guy was not flying, not debugging the autopilot, and did see the trim move because he has no instruments to scan and the trim is right in front of his face (in the middle between the two pilots). If you conclude that the issue is the trim, then any pilot would decide to go for the cut-out switches. It's just that concluding the trim is the issue, without knowing MCAS, under high pressure and with the wrong (autopilot) conclusion already on your mind is not very likely.