Short Answer:
Since all the weapons in the pack are fictional, most of them have a "serial number" or weapon name, i.e. Z27-A and a "calling name", i.e. the Fang, to make them more memorable.
I.e. "Hey, I really like the Prisma" is just easier for most people than "Hey, I really like the C-CS Particle Rifle"

They do have factions and the "white bone weapons" are usually referred to as Celestial weapons, since that is the faction that originally made them.

(really) Long Answer:
If you wanted to categorize the weapons by manufacturer, you'd go by:

1. HeatWave, produces mainly energy weapons. Most if not all of their weapons have a HS prefix.
   
2. Calhan, who specialize in reverse-engineered alien technology and did cooperate with HeatWave at one point. Their weapons have a three-letter prefix that usually starts with an A. They also did alternative versions of two HS designs, which now appear black and blue instead of metallic-gray and orange.
   
3. Serrath, with MTM and RK who, while independent in their design, have been born from the same faction and are usually good with heavy weapons, RK mostly focusing on traditional designs, whereas MTM is going full sci-fi.
   
4. Trinitatis with their T3i and TCTG lineups which specialize in elemental weaponry.
   
5. Union Armaments, HeatWave's competitor, who produces weapons with a Z- or VK- prefix.
   
6. Celestial, which will always have a C-** prefix indicating their origin and the class they were originally designated, i.e. C-AZ = Celestial, Ascendant Zealot.
   
7. Inferiant, the Celestial's arch-nemesis. Follows the same naming convention as the Celestials, but with an "I" instead of a "C". Note, that the Corruptor is technically also an Inferiant weapon, but still bears its RK prefix, since it's basically a corrupted RK56 revolver.
   
8. Helios, or rather "Order of Helios Knights"... which doesn't really make much sense to anyone else, since it's just one weapon, but that's the faction.
   
9. Miscellaneous. Some weapons just don't really have a faction they're coming from or are otherwise strongly affiliated or associated with. The Trace revolver would be such an example, but so are Hornet, Wasp and Stinger.

There are some weapons that just won't fit any of these factions and most of the archetypical designs come from the writing I did just to get inspiration going.
Truth is, some weapons just have more thought behind them than others. While the Celestial weapons have elaborate lore as a foundation of their design, the Hornet is basically "what if it shoots crystals that explode".

Ultimately, like I said above, their calling names are what they are meant to be referred by.
Since their backstory is usually just a creative tool for me, lore to some and trivia if not nonexistent for others, anything will do.

引用自 Darken217：

-Very Long Reply-

Thanks Very Much For Replying! I Was Getting Scared You Wouldn't, Given The Time This Reply Took. One Last Question, I Know You've Given The Manufacturers, But What Like.. Idk "Country" Or "Faction" Do They Belong To Individually? Thanks.

引用自 Czechoslovakia The Pencil King：

Thanks Very Much For Replying! I Was Getting Scared You Wouldn't, Given The Time This Reply Took. One Last Question, I Know You've Given The Manufacturers, But What Like.. Idk "Country" Or "Faction" Do They Belong To Individually? Thanks.

I usually don't check for new discussions, as I don't really get notifications when somebody creates a new one, unlike for posts in pinned discussions or the main comment section.

Generally speaking, there are 5 factions, 3 of which are human in origin:
The Calhan, the Serrath and scattered outlaws and bandit clans.
Alien factions are the Celestials and the Inferiant.

If you're so inclined, you can read up on the lore here:
https://docs.google.com/document/d/1tkuS9-yYKkSP-GOxMFq1ltIvVUW_euA-ojohcZlQIq4

While yes this is 3 years later but, not going to like the weapon manufacturer names remind me something out of Borderlands, which is really cool.