Are the Paradox Pokémon Ultra Beasts?
No seriously, the Paradox Pokémon have a ton in common, such as being a subgroup of Pokémon that aren’t regular Pokemon but aren’t Legendary and Mythical, have a shared connection to the cover legendaries, have a theme around numbers, abilities based around stat boosts, closely tied to the Main Antagonist, and are extra dimensional Pokemon with the Legendary Pokémon being physical/special swaps of each other.
With all those similarities, what if the Paradox Pokémon are just another kind of UB? Each of the UB came from another dimension, alternate timelines can be seen as other dimensions, so perhaps the Paradox Pokémon came from ancient/future versions of the Pokémon world; after all Guzzlord came from a distorted version of Hau’oli city.
They definitely share some parallelisms which you stated, but that aside, lore-wise they aren't the same thing.
I have seen some people arguing that given how according to what Professor Sada/Turo says in the Crystal Pool in The Indigo Disk, they would be Ultra Beasts by technicality given how they would supposedly come from other timelines and alternate universes (brought in to this one by Terapagos). And, by the way, that's what Ultrabeasts are: creatures from different universes which can be accessed through the Ultraspace (the world that connects every other world).
HOWEVER,
I don't think what Sada/Turo says is 100% real. I'll explain myself. First, this Sada/Turo's theory denies what their AI versions say in 'The Way Home', as they stated Paradoxes came from the past or the future from THIS timeline. This results in a contradiction which shouldn't be overviewed - yet most of the fandom has moved on and believed the past Sada/Turo's theory just because it's over there at The Indigo Disk and serves as a conclusion to SV's storyline.
Now I'll try to explain what I think Paradoxes are, because if we were to believe Sada/Turo, then Paradoxes are a kind of Ultrabeast indeed because of the Ultrabeast's definition I wrote above. This might be just a guess or something intuitive, but I believe Paradoxes are what the Scarlet/Violet's Book is in terms of what happens at the end of The Indigo Disk with the said book: as you may have noticed, Sada/Turo gives it to us, thus making the events of SV's original game impossible as there would not be any Scarlet/Violet book to look for the Herba Mysticas to heal Mabosstiff and the Time Machine wouldn't have had a way to be stopped. Not only that, but we know that the Time Machine needs to be created with the Briar's book (which we give to the professor in exchange for the Scarlet/Violet Book). This whole thing is, well, a paradox.
Now think of Koraidon and Miraidon's names, which are also a paradox because we teach this name to the professor and yet we knew this name because of the professor. I think this is a metaphor to explain what Paradox mons are, not only their names not the weird thing about the book and SV's story existence, but their origins as well: Paradoxes are just mons which shouldn't exist and yet they do, they have NO real origin, they just are generated by Terapagos' teracrystals at a certain point of our timeline (when the first Area Zero expedition came out) and then they disappear because they are brought to the present by the time machine.
EXTRA BONUS:
IDK if you have heard about this story, I heard about it and I think this comes from some series in Netflix or HBO but I can't remember where exactly. In this story, a famous musician well known because of their Mozart's interpretations has got access to a time machine and they decide to travel back to when Mozart started his career, in order to befriend him and congratulate him because of their work. Once they get back in time, they try to look for Mozart but they don't find him. That really pissed them off but they still keep on looking for him - and yet they don't archieve it. Eventually, the guy needs to work in order to earn money and subsist, what makes them show the world the pieces of art Mozart had composed. They end up being known as Mozart themself and becomes famous, so this results in a paradox. However, the most disturbing thing is: where did the music sheets come from? This guy had learnt about them at the school in the future; yet, they brought them to the past and used them to become famous. Who wrote the music sheets? - well, this is what happens with Koraidon and Miraidon's names in The Indigo Disk. And this is what I think Paradoxes' origins are too.