Maybe it was a coat closet back when people hung coats on hooks. It'd make a great area to store gift wrap or furnace filters. Hats could be stored on clips.

It was undoubtedly a closet for hats and coats. I would put brooms in it or remove the door.

I'd remove the door and use it as a coat/hat rack and put an umbrella stand in the little recess.

remove door & fashion a built-in shelving unit, or if that would look out of place (we need a photo of entire wall, or room), then fill it in