Is a manufacturer's serial number different from a US Government property number? For example, a military sidearm from 1918 might have a stamped number, related to the military ownership of that item.
Usually the US military assigns a serial number range to the manufacturer, particularly when more than one source of production is used, to ensure there are no duplicate numbers, and then uses the "manufacturer's" serial number as the property number.
If other stamped numbers or rack numbers are added at the unit level, they do not serve as the serial number.
You'll often encounter police and military weapons that are also stamped with a property number.
Fro example I have a Walther PP stamped with JW - short for "Justiz Wein" - which signifies the Vienna state police. You'll also find Austrian police PPs marked "SW" for Sicherheitswache Wien which is more or less Safety/Security Guard Vienna Austria.
In this case it is still just a property number, even though in this case the FFL had it listed as the serial number (and had to log it back in properly so he could log it out properly).
That happened because a) the FFL wasn't at all familiar with PPs and b) because the aftermarket grips obscured the serial number on the frame, which is located under the projection on the grip behind the trigger.
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However, it can get interesting when dealing with some foreign military and police contract firearms.
While the Vienna state police bought their pistols out of regular production, in some cases police departments in Europe bought them under a specific contract with a specified serial number range. For example a friend of mine has a Browning Hi Power with serial number "27". Given the actual age and configuration of his GP 35 (it's early, but not that early) it's clearly not the 27th GP 35 made. It is however the 27th example made under that particular French municipal police department contract.
As such that's a potential problem as there are potentially multiple Browning Hi Powers in the US using "27" for a serial number. That's one of the reasons importers have to either get a variance (which would not have bene granted in this case) or stamp their own serial numbers on imported firearms to ensure that each firearm of a particular manufacture and model has a unique serial number. In this case it is not stamped with an additional number as it's been in the US prior to 1968 when a serial number was required.
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In that regard, if a firearm was made for the US military, exported as military assistance, and then re-imported, a variance will normally be granted as the original serial on the receiver is unique - and that's desirable as it keeps the import marks to a minimum - but even then the import mark has to be visible.
For example, Big Sky re-imported a large number of M1 Garands in the 1990s and they had to have an import stamp. However, Big Sky put them on the barrel under the hand guard where they were not visible. That made the buyers happy, but ATF complained and they had to start marking them where it was "conspicuous" to clearly show it had been "imported", which was frankly pretty stupid as these M1s all originated in the US anyway.
You'll find import marks on a variety of US made arms that have been exported and then re-imported and it's important to understand that if for some reason the importer applied a new serial number rather than getting a variance, that's now the legal serial number, not the original manufacturer's serial number.