the picture definitely helps. It's nothing like the connector I was thinking about. I think you are correct that the plug goes to the brown piece and the brown piece to the board. from the look the two parts are just stuck together with years of grit and grime.
I would start with finding a way to clean all the dirt off so you can see what yo are dealing with. An air compressor, a can of computer duster, even a strong stream of water should help. If you have compressed air try blasting it directly into the joint perimeter to break the years of crud between he two halves.
You might be able to loosen it by GENTLY taping on the plug to seat it further into the socket. Then try to unplug it. If it was mine I would try a combination of WD40 and a stream of compressed air to force the WD into the joint.
Maybe try to use a channel-lok pliers laying sideways to grab the top and bottom of the plug and wiggle it loose.
Once you get it unplugged it looks like there are little tabs top and bottom. Depress these tabs to allow the brown socket to drop back through the metal plate.