Parts, having a serial number have zero effect on whether or not they can be shipped in the United States. Either it’s a frame/receiver, which is a firearm or it’s not.
As discussed, the Europeans currently have regulations treating barrels and both halves of upper/lower receivers as a Firearms but that has zero affect in the United States.
On older guns, multiple parts would often be serialized to be able to keep them together because the parts were not always 100% drop-in interchangeable.