I’m not sold on drywall since the top where it meets the floor could create a problem. I would remove what is there and finish it with wood, a nice stain to accent the light floor maybe.

Also try measuring between the 2 studs your finding, should be 16 inches. If it’s not, your most likely finding something other than studs. If they are 16, read that first comment again.

The metal hooks used to attach to a metal bracket that mounts to the wall first. Then it essentially hangs on that bracket. If this were mine, I would remove the back and metal hangers. Replace the back with plywood or similar to match the thickness of the top, then screw into studs through the plywood.