What is this door furnishing called?
It is a drip edge, it is usually applied on the outside face of a door that swings out. I have seen them on inswing doors too

There is no specific piece of molding for it in wood, but you can buy a metal door bottom with a sweep that has one integrated in with it. The ones of wood are simply a piece of molding or even flat stock with a sloped top attached to the face of the door at the bottom to direct the rain water over the threshold better