What is the structure from which a sign is hung called?
The wooden ones in these pictures are described as (ranch) arches, archways, entrances and gates, even when there is no moveable gate to close off the entrance. Most of them in the illustrations display the place name.
I'd go for...
gantry (Collins Dictionary)
A gantry is a high metal structure that supports a set of road signs, railway signals, or other equipment.
The fact that that particular definition says a metal structure isn't significant to me. Functionally it's the same thing if it's made of wood.