Back in the day, this was done by mail order houses to differentiate between different ads.
I don't believe it. How would you even know this? Unless you can actually show evidence for this absurd claim, it's pretty obvious you just made this up. What reason would they have to differentiate between ads?
Because this ground has been trod over many, many times before over the decades. If you don't want to believe it, that's not my problem. But, really, I've actually had good acquaintance with people who perform what is now called "analytics" for pre-ecommerce catalog sales, and found out about this independently of my interest in the JFKA. Big ads in national publications weren't cheap, so the advertiser needed a way to know which ads brought in orders and which did not. This, in turn, influenced future advertising buys.
By law, the serial number of the firearm had to be carefully recorded when it was sold.
True, and not only that. There needed to be a registration of any weapon being sent in the mail. So, where is that paperwork for the rifle?
Nope. The only firearms that would need to be registered were/are NFA items like short-barreled (i.e. <16") rifles, machine guns, and silenced weapons. Carcanos did not fall under any of these categories.
But not for the ad's "department" number (assuming that's what it is). The department number, if that's indeed what it is, is randomly squeezed into some free space between two lines of text in another section of the order form. It's not associated with the line item representing the rifle.
More selfserving flawed made up nonsense.
If it was that flawed, you'd be able to point out the flaws in my statement.