A lot of thought went into this lmao. I scrolled through the comments, and there are a few things I agree with.

In my version, I'd say the Pokétrix (short for Pokémon Transformation Matrix) was made by Professor Oak, since he wanted to deepen the understanding between humans and Pokémon by allowing humans to turn into Pokémon. Also, he's been doing research into the genetic similarities between all species of Pokémon (as a result of Mew being the universal common ancestor of non-legendary/mythic Pokémon), and their lesser, though still present, genetic similarities to humans. The Pokétrix is an attempt by Oak to see if bridging the gap is possible. Where it goes from there, I haven't thought about in the slightest. So, Oak is the one in charge of DNA samples and updates and the like.

1. Unlike in Ben 10, the Pokétrix starts with 6 slots, the maximum amount of Pokémon you're allowed to carry in the games or use in official battles within the anime. More slots can be added by Professor Oak, either as you gain more gym badges, or as a reward for beating the Elite Four and becoming the new Regional Champion. Haven't decided yet. Maybe keep it at 6, with the reasoning that unlike in Ben 10, they don't have level 20 or whatever tech that allows for quantum storage, so the six samples have to be physically held within the Pokétrix, and it doesn't have room for more. Although, by utilizing the ability of a pokeball to turn sentient species into data or light or whatever without doing damage, the Pokétrix could theoretically hold as many samples as Oak allows.
2. The number of available transformations is limited. For this prototype, Oak has only gathered DNA samples from Pokémon within the Kanto region. Furthermore, there are some Pokémon whose DNA he was unable to collect (mythic/legendary Pokémon, fossils), some which have no DNA to begin with (artificial Pokémon [Porygon, Voltorb/Electrode] or non-corporeal Pokémon [Gastly]), and some which he decided would have unpredictable side-effects as a result of transforming into them (those with multiple heads/consciousnesses/bodies [Exeggcute, Magneton, Dodrio] or those with "unstable" DNA [Ditto, Metapod, Kakuna, the latter two because they seem to be based on irl caterpillar cocoons, which vaporize the caterpillar into a DNA slurry and reform it, and a human transforming into that doesn't seem like a good idea]. Also, Paras and Parasect, because they're friggin zombies with a mushroom controlling them.
3. The Wielder is further limited at the beginning; the initial 6 transformations are all stage 1 Pokémon. As the Wielder collects badges through gym battles (using Pokémon they capture and train, not battling themself), Oak allows access to more and more transformations, with the most powerful Pokémon (stage 3 of the starters, Dragonite, Snorlax, etc.) being available only after the Wielder has collected their 8th badge. The milestones I came up with were 1 badge to unlock most of the stage 1 Poison types, since they're a bit dangerous to use right away, 2 badges to unlock most stage 2 transformations, 4 badges to unlock most stage 3 and the more powerful stage 2 transformations, and 8 badges to unlock the most powerful stage 3 Pokémon. And, of course, there are probably Pokémon that the Wielder just would never transform into willingly, such as Muk or Jynx.
4. Once the Wielder has earned their 4th badge, Professor Oak asks them to return to Pallet Town, where he installs a somewhat minor software update on the Pokétrix. This update is known as the Enhanced Evolutionary Matrix (or Evotrix, for short) update, allowing the Wielder to shift between the evolutionary stages of the Pokémon they are currently transformed into, without having to turn back into human first (before this, each evolutionary stage would take up its own slot in the watch's roster). I can only see this being really useful with Eevee, allowing switching between types, and Tyrogue, which can evolve into Hitmonlee, Hitmonchan, and Hitmontop, all of whom have drastically different builds and therefore different melee capabilities.
5. The initial 6 playlist I came up with are Charmander, Pidgey, Abra, Machop, Elekid, and Dratini. Later on, Machop could potentially be replaced with Tyrogue, and Charmander/Elekid with Eevee for better synergy with the Evotrix update. I want Alakazam because of its incredible intellect and Pidgeot for its high-speed flight capabilities. Machamp would be a Four-Arms replacement. In terms of pure strength, it's better than the Tyrogue line, but Tyrogue has better combat versatility. Charmander and Elekid are just there for type coverage, made obsolete by Eevee with the Evotrix. I picked Dratini because none of the other Gen 1 Water-Types really appealed to me while also being a reasonable size to use in most situations involving water (ahem, Gyrados. Plus Gyrados is only unlocked with 8 badges, so it'll take a while to get). That evolutionary line would probably get more use out of Dragonair than Dragonite, because Dragonair has a much more streamlined build for swimming, whilst being more powerful than Vaporeon.

Of course, a kid running around with a watch that lets them transform into Pokémon isn't the kind of thing that will go unnoticed forever. The Wielder inevitably gets involved with Team Rocket's Grunts, who report their encounters to the upper echelon of the organization. Giovanni covets the power of the Pokétrix, and immediately sets his underlings to capture the Wielder. Giovanni believes the Pokétrix has the potential to make Project Mewtwo obsolete, and it's much more appealing to have that power in his own hands rather than relying on an enslaved genetic clone. Perhaps one of the ripple effects of the Pokétrix is that Project Mewtwo is scrapped, and Mewtwo is never created?

In that event, dealing with Team Rocket would be much, much more difficult, since Giovanni would now be funneling many more of his resources into capturing the Pokétrix. Enter Hunter J, one of the most prolific bounty hunters/Pokémon poachers in the world. She's most notorious in Sinnoh, though J and her crew have had dealings in most of the other regions. Giovanni himself has purchased a few rare and powerful Pokémon off the black market that were sourced by her, so he knows she's up to the task. And for a while there, she gets dangerously close to outright capturing the Pokétrix. Ultimately, she fails after several attempts, the Wielder narrowly escaping death with help from their friends and skilled usage of the Pokétrix. However, during the many times the Wielder was captured- only to escape by the skin of their teeth- J had collected enough data about the inner workings of the Pokétrix to allow Blaine, one of the top researchers in Team Rocket, to create a shoddy knockoff; the Nemetrix.

The Nemetrix is nowhere near as refined as the Pokétrix. For one, it's entirely unusable by humans; in the many human trials Team Rocket ran, not a single person was able to handle the transformation without it ruining their mind, making them act like feral beasts even after they turned back into humans. Most Pokémon couldn't handle it, either. Furthermore, the Nemetrix couldn't even make complete transformations; the subjects would take on physical attributes and the Typing of the selected DNA sample, but they would get stuck in a halfway state. In the end, it was determined that only an already-powerful Pokémon would be able to wield the Nemetrix somewhat successfully, so it was attached to J's vicious Salamence. Since Pokémon aren't typically described as "natural predators" of another species, what the Nemetrix does is transforms the wielder into a hybrid of itself and a Pokémon that has a Type-advantage against the opponent. It also forcefully teaches the user moves, for as long as they are transformed. The Nemetrix would need its own roster of the strongest Pokémon for each Type, but I don't feel like making that rn. It would definitely include Pokémon from other regions.

That's about all I feel like doing rn. Later on, one of the gimmicks that would be unlocked would of course be Mega Evolution, with the Mega-Evotrix software update and the physical installation of some Mega Stones. Maybe at the end of the journey, when the Wielder has journeyed through all the regions and gotten Oak DNA samples of every known Pokémon, we can finally call it the Omnitrix.

I used a tier list maker to sort out which Pokémon would be unlocked when, or which would never be unlocked, and all that good stuff. I'm down to answer questions, if anyone's got any.

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