I understand how to do pinch harmonics (not that I'm particularly good at it) but I don't understand why they work. In particular, why do you not have to touch the string at the exactly 12 frets up from where your left hand is fretting?

Secondly, and related to this, I am aware that there are places on the string that are generally better for getting pinch harmonics (apparently independent of what notes you are playing / what frets you are fretting). Why is this? Is it related to where you are pinching in relation to the pickup? In this instructional video, the sound really changes a lot depending on where he's pinching. Are these all the same notes just with different octaves emphasized? I assume they must be, right?