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That's called a "support". Looks like PN 997-537-331-01-01C. It mounts to the door as you suspect. Not the car.
I'm not sure how it mounts. Online parts diagrams don't show any separate mounting parts. Perhaps it's glued on, but the way the holes line up makes me think there's a mounting piece that might have broken and fallen down into the door. Got any new rattles in the door?
They're cheap if you need to buy a new one. About $30.