|| since rabid spoke about it, lemme share some of my thoughts on
How the pokémon Ho-Oh is not the same as the Deity Ho-Oh;
Just like the pokémon Arceus isn't the Deity Arceus, same thing for Dialga, Palkia, Giratina, etc;
What it means is: the pokémon is merely a "hand" or "avatar" of the Deity. Like a physical, tangible avatar of its will to allow it to be present and ever observing the mortal/tangible world where humanity and pokémon live.
So when I say Ethan has Ho-oh in a pokeball, it means the Deity Ho-Oh has allowed its chosen, Ethan, to keep its avatar with himself. Trusting he'll be a good guardian and he'll "use" its avatar to good and justice.
Which is why there are more than one Ho-Oh canonically (as it also shows up in Colosseum, on the opposite side of the world). There may exist several avatars/hands of the deity, in the same timeline, or in different timelines, and such is not odd.
As for the Deity, it is a single, unique yet omnipresent/intangible Deity across every timeline and every world. A deity, therefore humans are unable to fully recognize or dimension its real nature or existence.... Which makes the avatars/hand a necessary measure.
It's important to note this is about Legendary Pokémon who are Deities. Mewtwo, for example, is not an avatar. Neither is Koraidon, nor Calyrex, nor Spectrier, etc.