(besides covered in an unholy amount of paint)
This mystery bit of hardware is attached to the chair rail in our tiny bathroom. It never occurred to me, until now, to wonder why there is chair rail in our 27 square foot bathroom. Maybe to top off long-gone bead board? The location of the bathroom is original to the house, and the chair rail matches the type in the kitchen and dining room.
Anyway, this strange object is attached to both sides of the pocket door (one by the toilet side, one by the sink side).
It looks like it’s meant to hold something, but what? Can’t be a towel hook, because it would make the towel hang almost in the toilet.
I spent way too much time poking around online vintage plumbing suppliers to see if I could find a fixture that looked like it might fit. I didn’t find anything, however, I did come across the worst plumbing-related invention ever. Behold, the folding urinal:

from Vintage Plumbing
The Victorians never cease to amaze.