I don't know if they make some kind of record in the system or not (you have to wait outside most of the time), and if so what. I'm guessing they just make a note on the visa in the old passport and stamp another one in the new one. There's an extra small stamp that says "สำเนา (copy, duplicate)" in Thai above my extension of stay in the new passport. Hope that helps.
On the basement of Promenada, there's a few small rooms with glass doors to the left of the main office, that's where you need to go. They'll check your docs at the counter desk to the front. It's free and it took me 1 to 2 hrs max. You need to get the letter from your embassy first though. Good luck 😊