Neither of these fit my answer. They aren't separate universes. At least when thinking about the original vision for Paper Mario.
The "Paper World" as a separate universe exclusively appears in Paper Jam, which I struggle to find reason to even recognize as canon to the M&L continuity given just how much is randomly super sanitized for just that one game. Outside that game M&L has a strong sense of continuity with multiple recurring original characters and/or species between any two games. PJ explicitly abandons all of that except for Starlow, who alone just isn't quite enough (and has like no explanation for why she's present unlike in DT where they establish that she's vacationing)
So now that PJ and my general distaste for that game as a definitive source is addressed, yeah the original Paper Mario was conceived as a sequel to Super Mario RPG, with its distinct visuals being to convey a storybook style to fit the tone of the story. They were never meant to imply canonical things about the world that splits it off from the main Mario world.
Its first two sequels make heavy use of species like Clefts & Whacka and races like the Shaman clan (who date back to Super Mario RPG, but their specific named members were introduced in Paper Mario). They do incorporate the art style into gameplay more, but I don't believe this causes the stories to be incompatible with happening in the core Mario world. It's not hard to visualize versions of the curse transformations being applied to a non-stylized Mario, he merely becomes flat temporarily via magic. I made a visualization of how this could look last year:
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When it comes to the 3D flip from SPM, all you have to do is imagine the base 2D dimension from SPM as being exactly how any 2D Mario game is. The same world that exists in 3D, just being shown to the player in 2D for gameplay purposes. The fact that the intro features two locations from TTYD at the 2D perspective supports this. With this in mind, it changes nothing narratively about the ability Mario gains, it's just globally dialed back a dimension so the player can perceive it. He is granted this ability magically by an abstract interdimensional wizard, so in-universe he could easily be transcending to a higher dimension that we wouldn't be able to perceive.
There's also the appearances of classic Paper Mario characters in some 3D games in the 2000s. Something I wish could happen again but sadly since Nintendo shifted their framing of the Paper Mario series the chances of that have gotten as bleak as ever.
As for the modern trilogy, yeah i don't have much backing them up. They are pretty overt in their insistence that everything is actually physically crafted from parchment in-universe, with paper puns literally being inserted into the dialogue for moments that in my view would have included a laugh track if acted out live. Other than the way the characters are drawn, almost nothing about these games feels like it has any interest in building on the continuity established throughout the first 3 games, and instead build exclusively on a rebooted foundation established in Sticker Star (with all species and characters debuting in the first 3 games scrubbed from existence). So yeah these i can much more easily picture as being part of an alternate universe where everything is actually made of paper for reals.
That said I do think VERSIONS of these events happened in the core Mario world. They'd just look way different. Instead of Royal Stickers there'd be Cosmic Crowns or something, Just like the Star Spirits, Crystal Stars and Pure Hearts had nothing to do with paper and were just straight up magical entities organically built into the world. Color Splash could still have its paint shenanigans but it would be handled more like Mario Sunshine or Bowser's Fury rather than being something that directly plays on paper theming. And King Olly could still be an origami villain, as I could still see origami being a hobby of non-paper beings in the core Mario world. Instead of folding paper people into origami, he uses magic to flatten people and THEN turn them into origami. His minions would be other origami creatures folded to life by him, rather than a bunch of sentient art supplies that only exist to play off the idea of a paper world. They could still wield some of these items, like scissors being wielded like swords as that would still be a threat to an organic Mario. And their personalities could remain intact. Bottom line is, there exist hypothetical versions of these stories that could feasibly take place in the main timeline in the main universe, they just aren't the ones we see.
TL
R: The first 3 games absolutely happened in THE Mario world, and that is the version we see. The "Paper Mario Style" is a stylistic choice for presenting it to the player, and all paper elements in gameplay are explainable via magic. The modern trilogy as we see them are most likely in a separate world, possibly the one from Paper Jam but i really don't care enough about Paper Jam to definitively decide on that. But their events in some form or another most likely did happen in the main world, we just didn't see it.
