Legendary Bird: Multiple.
Mewtwo: I actually wouldn't be surprised if there was a few. Mr. Fuji couldn't create Mewtwo by himself, and in the games it seems like all the scientists lived, and this seemed like a pretty sketchy experiment which may have meant not all the scientists were just in it for the science or good of the world. All it takes is making a copy of the blueprints and selling it to the right contacts on the black market.
Mew: Multiple.
Legendary Beasts: Despite sounding like they should be one-of-a-kind, there are multiple. Seems like the Ho-Oh species came across a winning formula and seemed to have created multiple trios over time; which means for every Legendary Beast trio there's a tragedy connected to them where at least three Pokemon died.
Tower Duo: I believe there are multiple (let's put aside the anime, we're talking about the games). The ones in Johto are just the most well known as they were worshipped.
Celebi: So its either there are multiple, its the same Celebi but at different points throughout its possibly eternal life, or a combination of the two (like there's multiple timelines which at points cross with one another and each timeline has a Celebi).
Legendary Titans: Multiple.
Eon Duo: Multiple.
Weather Trio: Hard to say, like it would make sense for them to be one-of-a-kind. If there are multiple, at the very least only one group of Trio is active at a time, it's just that the ones in Hoenn are the most well known cause their territory just so happened to be a populated region.
Jirachi: I say multiple, the thing is most are asleep and are either disguised or slip into a pocket dimension until they wake up. Heck, I always question its Dex entry and came up with a theory that only ones which have granted wishes during the seven days is at its strongest need to sleep for a thousand years. All others have a normal sleep cycle but are unable to grant wishes (at least major ones, at most maybe able to manipulate luck) when it's not the seven days.
Deoxys: Multiple, but most are likely still in a dormant state.
Lake Guardians: Ugh, hard call. Like, lorewise they would be one-of-a-kind, but the in B2W2 threw us a strange curveball with the Cave of Being which is apparently connected to Sinnoh (I'm assuming not literally) and the Lake Guardians. Did Arceus place multiple Lake Guardians around the world to make spreading of emotions, knowledge, and willpower quicker? Or is the Cave a Being a place the Lake Guardians can directly teleport themselves to and there's a few places like it around the world?
Creation Trio: Well, if we go by events in the game, they all used to be one-of-a-kind but then Arceus up and goes and creates an additional one using the Mystri Stage in the Sinjoh Temple. Now, since Arceus created that additional one for the player, its questionable whether its considered "active" or, as long as its with the player, it's considered "inactive". Also what would happen to it when finally set free? I guess that's another question entirely.
Lunar Duo: Multiple.
Heatran: Multiple.
Regigigas: There does seem to be multiple.
Manaphy Family: Multiple.
Shaymin: Multiple.
Arceus: That depends what you're talking about. If you're talking about the entity, one-of-a-kind of course. If you're talking about the physical body we know as Arceus, well Legends: Arceus proved a theory I had that it's merely just an avatar created by Arceus to interact with the physical world and it can (and probably has) created multiple.
Victini: Likely multiple, though nothing really hints to it, but I don't see why there wouldn't be.
Swords of Justice: Multiple.
Forces of Nature: Probably multiple?
Tao Trio: From lore alone would be one-of-a-kind, if anything there could be multiple of the Original Dragon. Unless the splitting of one caused a ripple through reality which caused all Original Dragons to split. Problem is they don't really go into where the Original Dragon came from, it just exists in the legend and then splits to create Zekrom, Reshiram, and Kyurem.
Meloetta: Probably multiple.
Genesect: Anime aside, the issue with Genesect isn't if there are multiple, is if multiple have been made. If there's likely more Mewtwo and we know there's more Type: Null, then I would say the chances are good that a former Plasma scientist got the blueprints onto the black market where some fossils of the ancient Genesect have also likely shown up.
Aura Trio: Seemingly one-of-a-kind? According to Dexio & Sina, the Zygarde that appears in Alola is the one from Kalos, having come to Alola because it sensed a disturbance (the Ultra Beasts). Though there could also just be that most of the time Xerneas and Yveltal are inactive. Also Zygarde likely can create new cells & cores if something happens to one of them, and if it can do that than another Zygarde doesn't seem farfetched (if I had to guess, they would share a mental link so would just be two of the same; no need to have multiple then unless it's going into a dangerous situation and leaves some Cores and Cells behind in case the worse happens).
Diancie: Multiple.
Hoopa: Probably multiple?
Volcanion: Despite seeming mechanical, it looks to be organic so see no reason they wouldn't be multiple.
Type: Null Family: Multiple, not just because Aether Foundation made three but also because it's revealed in SwSh that the blueprints are on the black market.
Guardian Deities: One-of-a-kind.
Cosmog Family: Multiple.
Necrozma: USUM certainly makes it out to be one-of-a-kind.
Magearna: While another artificial Pokemon, the Soul-Heart is a really notable part of it as that doesn't seem like something you can "just" build. If multiple do exist, its just a few, and who knows if more can be even built (its way too old for any blueprints, at least in good quality, of it to still exist; unless the Soul-Heart is that complex to build).
Marshadow: Probably multiple?
Zeraora: Probably multiple?
Meltan Family: Multiple.
Hero Duo: I'm guessing one-of-a-kind at this current moment.
Eternatus: One-of-a-kind on the Pokemon World, who knows if or how many are flying around the Pokemon universe.
Kubfu Family: Multiple.
Zarude: Multiple.
Calyrex: One-of-a-kind.
Calyrex Horses: Like, if Calyrex is one-of-a-kind, it seems like his horses are too?
... Also multiple dimensions (Hoopa's rings, Ultra Space, possibly Dynamax Dens/Lairs) means there are plenty of meta explanation why we're running into Legendaries that are one-of-a-kind or there's very few of.