The situation couldn't be simpler.
"shop door" is a: type of door.
That would be because, you know, shop is an adjective? :)
Shop door, cheap door, expensive door, pink door, last door, aircraft door, submarine door, waterproof door ... shop door.
Adjective-noun. Couldn't be simpler.
the shop ’s door is: the door of the shop you are referring to.
shop door, pink door, cheap door: type of door
John's door, the shop's door, the barn's door: doors belonging to John and others
Again, couldn't be simpler.
Consider say "castle gate". That's the highly decorated, large, baroque, type of gate. (Note, type of gate.)
It would be completely normal to say:
"This castle's castle gate has gold accents, but this castle's castle gate is rusty."
You could also say:
"Here's the castle's castle gate - and over there is the castle's small gate, and over there is the castle's emergency gate."
One point of confusion: "shop door" (i.e. adjective-noun) is an extremely common term. So when you say "The shop's door" it just, happens, to sound similar to that extremely common phrase. No big mystery.