One license for "possession of firearms" or even one generic "license to carry" makes sense for streamlining play.
It's hard to generalize, given that weapons possession/carry laws vary widely between jurisdictions, both in RL and in-game. RL example: In Texas, you don't need a license to own a firearm (aside from the federal background check that occurs when you buy it) and you only need a license for carry. In New York, you register each individual firearm, and good luck getting a license to carry at all unless you're a LEO. Given that difference, one could justify having the firearms laws vary between NYC (SRM 03) and Seattle (SRM 04).
Even driver's licenses vary in RL by what class of vehicle you're allowed to drive. A passenger car driver's license does not permit you to operate a tractor-trailer, for example, nor does it necessarily even permit you to drive a motorcycle.
But yes, there's nothing served except administrivia by requiring characters to list their 37 fake licenses for each specific weapon, activity, etc.. It should be enough to specify "weapons", "magic use", or other broad category, noting that generally nothing with an availability of F can be licensed.
Although technically you shouldn't have the same firearm licensed under more than one SIN. I don't think it's really spelled out, but you run into the same issue as if the cops run your prints and two SINs come up...even worse if that firearm is known to have been used in a crime.