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Like, is the Mudsdale you ride on Hapu's Mudsdale, or is it a different mudsdale? It's never made completely clear in the game, but I assume they aren't the same Pokemon because not only would that not make much sense, but there's a part of the game when you can ride Mudsdale right in front of Hapu's. And the one you ride is smaller than her's.

Hapu's grandmother also gives you a Machamp ride pager, and it seems like she suggests that it's the same Machamp you see with her. But, again, you can "ride" (be carried by) Machamp in front of her Machamp.

It confirms in the game that it is a different Pokemon. I could source that with a quote from the game but I'm lazy and I don't think I can get the quote back again now that I've finished the game idk.
You can tell me to get rid of this answer if you want, I don't mind.

#My Illogical Theory
(I haven't done one of these in so long omg)

 I'm certainly sure the Pokemon you call upon with the Ride Pager do not belong to the Trial Captains of the islands. I say this not only because of the points you listed above, but also because most of the Trial Captains and the Elite Four do not battle with anything that could be a Ride Pokemon; Tauros, Stoutland, Lapras, Sharpedo and Charizard are species that do not belong to any important characters in the storyline (this excludes Trainer Red at the Battle Tree and Youngster Tristan, a trainer who attempts to dethrone your Champion when you rematch your Pokemon League). Although, this can be contradicted by the idea that the Trial Captains or Elite Four wouldn't need to necessarily battle with those Pokemon.

There's also a quote in the game that specifically states that a registered Ride Pokemon would come to you when you call them. The quote was written to indicate that there existed many Ride Pokemon of the same species... I'm sorry I couldn't find it. If anyone is able to find it, please comment below!

Hope I helped. :)
Source: Respective Pokemon species' TrainerDex entries on Serebii.
Doesn't Hala say that the Tauros in your Ride Pager is his? Not that he battles with it, but it's more like his pet/work animal?
Hala and Hapu both make statements indicating it is their personal Pokemon that come to you, however, there is an encounter at Panolia Ranch that directly contradicts Hala.  The one where you have to battle the Tauros so the Miltank can calm it down, so the Pokemon Breeder can train it to be a Ride Pokemon.  And there's the fact that Charizard isn't even available in the Alola region, and yet it is a Ride Pokemon....  I like that HMs are gone, but it kind of feels like GF didn't tidy up their quick notes when they put the new system together.
I mean
Hapu's grandma-Machamp
Mallow-Stoutland (I think)
I'd like to bump this as it's currently still unresolved.

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