While this reuse of a serial number is more problematical with some guns more than others (the Luger is probably an example of the worst case), it's a problem to some degree with many firearms. In terms of shotguns, I recall reading an article where the current owner of a very nice English shotgun (Purdy? H&H? Memory fails.) wrote the maker for information on the gun. The information supplied didn't match his gun at all (even the gauge was wrong, IIRC). After much back and forth it was determined the maker had, for some long forgotten reason, made two guns with the same serial number.
Anyway, just for giggles (and this is seriously over-kill) here is the extent of the problem for Lugers.
Military Lugers were serial numbered in blocks of 9,999 pistols. Any given production run would start with serial number “1” and continue through serial number “9999”. The ten thousandth Luger in that production run would be numbered “1”, with a lower case letter “a” suffix. This numbering sequence would continue to serial number “9999 a”. The next 9,999 Lugers would be serial numbered “1 b” through “9999 b”, and so on through the alphabet. When the alphabet had been used, this would start all over again with the serial number “1”. Often new production runs were begun before reaching the end of the alphabet. Each German manufacturer of the Luger followed this same serial numbering methodology.
The bottom line to this any given four digit serial number was used over and over again. To have any hope of uniquely identifying a military Luger one must have the serial number, the serial number suffix (if any), the chamber marking (if any), the manufacturer, the barrel length, and the cartridge for which it is chambered
A German military Luger will have the full serial number and the letter suffix (if any) only on the frame, (just under the barrel) and the magazines (on the magazine base). It will have the full serial number, but not the suffix, on the barrel (on the bottom rear) and on the barrel extension (on the left, just behind where the barrel is screwed in). Most of the rest of the parts will have the last two digits of the serial number. When we say a Luger “matches” we mean all these numbers are the same.
Let’s say we have a 1917 DWM military four inch barrel, serial number “1234 a”.
DWM
1908: 1 pistol
1909: no pistols observed
1910: 1 pistol
1912: estimate production 10,000 or less, maybe no “a” block pistols
1913: 1 pistols
1914: 2 pistols
1915: 2 pistols
1916: 2 pistols
1917: 2 pistols
1918: 2 pistols
Erfurt
1912: 1 pistol
1913: 1 pistol
1914: 1 pistol
1915: No pistols reported
1916: 1 pistol
1917: 1 pistol
1918 1 pistol
Estimated total number of Lugers with four inch barrel, serial number 1234 a produced during Great War: 19
Luger production never really ended after the Great War, and continued sporadically until the early 1930s, when massive Luger production began again. Military production of the Luger stopped in Germany in 1942. I'm too lazy to look up the numbers of runs and guns, but guesstimate another 20 – 30 Lugers with four inch barrels with the serial number 1234 a.
So figure 40 – 50 guns SN 1234 a. Assume John Doe had one, doesn't have any idea about how Lugers were serial numbered, and had his Luger stolen. Five years later he sees 'his Luger', SN 1234 a, displayed on the internet. He calls the police, files a report complete with his old bill of sale for a Luger SN 1234 a, and we're off to the races.
Seriously ugly.