It's generally accepted that Pokemon is a multiverse. ORAS made it apparent when we were going to use the Link Cable to sent the meteor to another universe. However, after diving into some lore, I think each game itself is a multiverse and Pokemon as a whole is bigger than a Multiverse.
So when ORAS introduced the idea of multiverses, it made sense. And then we got Sun and Moon and, more importantly, Ultra Sun and Ultra Moon. Here, we are introduced to Ultra Space. And Ultra Space is considered to be interdimensional in and of itself. Heck, we're even able to travel to an 12-hour offset version of Alola. My copy of Ultra Moon is a multiverse in and of itself. But no matter how much I use Ultra Space, I cannot reach someone else's copy of USUM. My game is an isolated multiverse and I'd still have to use a "Link Cable" to connect to another game.
Bonus evidence: As far as I can tell, only one Necrozma exists across Ultra Space, which means only one exists in the Multiverse. But everyone can get their on Necrozma in their own games. Plus we also have to consider Rainbow Rocket pulled villains from different universes as well.
A Metaverse is the term used with describing a collection of Multiverse. So, Pokemon is a collection of similar multiverses at this point, forming a Metaverse between all our game copies, anime, manga, and so on and the Pokemon Hyperverse is a part of the larger Omniverse.
EDIT: Clarification on what a Metaverse is. To my understanding, a Metaverse "hosts" a number of Multiverses the same was a Multiverse "hosts" a number of Universes. Since Pokemon is comprised of a multitude of Multiverses, it should be considered the next rank up from Multiverse, whatever that word should actually be.
by vernanonix