It also can not be dropped or swapped for another item by drag-and-drop. The only way to put the shotgun away is to bind another item to a quickslot and press that key to switch. Then it will be put on your back.
It is also impossible to drag the shotgun to a quickslot or to the hands. The only way to get it into hands is to eject the ammo - then it moves to hands, with the ammo both in the backpack and visible in the shotgun.
There is no way to drop it on the ground.