What technique is used to cut these door moulding?
Testor101 is correct in both points. Coping this joint with the coping saw is the way it was most likely done when it was built. It does allow for a corner to be a bit of out of square, but doing this will be taxing on the blade of a coping saw, it is a lot of wood to cut through. But it can be done with patience. Trying to rush through a cut this size will break blades.

If you have access to a band saw, it would make it much easier to do, but to get it a tight fit would still require some tweaking. Whether you run the stock at a slight angle after the cut is started to give the cut an extra relief in the back, what I have been taught is "back cut", so the face of the cut is absolutely tight but the mating faces are gapped slightly, but that does not matter since it is not visible. You could use a rasp to tune the cut too. This back cut can be done with the coping saw too.