Meh, I just can't see it. An item is an item, going bananas with project names would just make things confusing.
I think it's telling that players keep commenting on this. I also agree with the others that it's all kinds of silly.
If an agent wants an axe because he's some kind of axe hero that fancies facing alien monsters with one or they want a katana or a machete... they can just go and buy one at a shop or online. You can put those things in checked luggage even as a civilian. And although you might get some weird looks, you can also walk around with one, especially if you put it in a shopping bag ("What? I just got this axe. I'm going to the cabin this weekend and I'll need to split wood for the campfire!"). Then when the monsters come out, or when you go in the middle of the woods to find them, who's going to argue that you shouldn't have saved that lady by hacking off the wendigo's arm with an axe?
The only point where it is a bit more iffy is when facing humans, since some could argue that as a legitimate organization, you shouldn't apprehend humans with axes and katanas. That's just not socially acceptable. So maybe those deployments prevent you from bringing those weapons because of ethical reasons (ex.: only proper, stunning melee weapons like stun batons and tonfas are authorized against humans?)? But having your team of researchers work for a few days on axes, then katanas, then whatever that other Japanese sword is, then machetes, etc. just is jarring. Either they're making a legal case for using lethal weapons in those operations, which should unlock all of them at once, or they're doing something that no 20th century researcher should have to do: Discover how an axe works.
I understand that firearms are controlled in certain countries and that you need to go through the various bureaucratic channels to enable your agents to use them. But the credibility of the story told in XComFiles would indeed benefit from some streamlining of modern stuff. Discovering axes is
not exciting, which imo research results should be. Even in an alternate bureaucratic world, it makes no sense so it hurts the suspension of disbelief by breaking the in universe "logic". Despite UFOs and aliens being less believable for us, they are one the those "basis of the world" that you accept when starting the story, so the "There's aliens and what you find unbelievable is researching axes?!" line really doesn't work either. If you make part of the story that the world in XComFiles is one where all dangerous objects are banned by governments (how do people split wood? You need an axe permit like we need a gun permit?), maybe.. but that seems rather unnecessary.
I would also personally prefer if the kind of stuff you can find in a hardware store (axe, machete) or order online (combat knife, katana, tonfa, all kinds of swords) were all in the pedia and shop from the beginning, or all unlocked together with a "Ethics and legality of using unusual weapons" project. The current situation detracts from the quality of the experience by taking you out of the story, thinking: "What? you really needed 5 days to tell me an axe is choppy?".