I think that BATF considers any part that bears the s/n to be a ”firearm”. I could be wrong - if so, someone will tell me.
The frame and receiver on a P.08 are two separate parts but, each carries the s/n. The frame is the “bottom half“ and the receiver, with the barrel screwed into it, is considered the “top half”. One can fire a P.08 without the frame attached.
A certain ”serial number” on a P.08 probably appears on a few hundred different pistols. In order to exactly describe an individual pistol you need the s/n, manufacturer and date.
For instance, take the s/n 1234. That number appears on a pistol from each of the manufacturers once a year, because each mfgr. started the numbers at 0 each year. So in a given year, “1234” appears on a pistol from each mfgr. Consider how many years each mfgr. made these and you have an idea of how many pistols carry that number. But. there is only one pistol that is identified as “DWM, no 1234 ,made in 1916”, only one pistol identified as “Mauser no 1234, made in 1937”, etc. K98ks are the same way. If you talk about a Mauser Oberndorf rifle with 1234 stamped on it, there are 13 of them like this, but if you specify the year, there is only one.