There is little restriction on pokemon types that are not legendary or mythical. At the end of the day, you can do whatever you want forever but I personally prefer to see pokemon characters that invoke at least one pokemon. If the character no longer looks like at least one or two specific pokemon anymore, it's just sorta some other critter, right? Consider playing with sizes, markings, hues, and so on. If your pokemon looks off-base model, consider if there is something in their lineage leading to that! (or maybe not!) Pokemon may be born with these variations, or find they obtain them in a variety of ways later on in life (consuming a specific food engineered to make the change, as a result of spending too much time in unpredictable mystery dungeons, etc. Get creative!).
While shininess is far more common in this setting, so are plenty of other colors. It's extremely common for pokemon to still be their base colors, or maybe shifted just a bit here or there, but it's not unheard of for pokemon to be completely different colors. Some socities may hold "canon" shiny colored pokemon in high value, calling back to their rarity long long ago.
With so much excess energy in the lands, some pokemon are finding they can wield types their species never could before. Pokemon still only have two types they can be aligned to (with an exception explained later on), but their original types can be replaced entirely. Their appearance will likely reflect these biological changes, but does not necessarily have to. Pokemon generally will not take on attributes associated with a specific type if they do not have that type themselves.
Occasionally when a pokemon is released from a pokeball, it may appear very different from its "standard." At some point long ago, humans were busy creating breeding lines for pokemon of interesting colors, patterns, and other miscellaneous traits. Think like an extra long-haired eevee, a glameow with a tabby pattern, or those funky magikarp from magikarp jump. These new traits could feasibly be introduced into the current day's breeding pool, should the right pokemon be released from its long slumber.
Pokemon don't always pop out looking like one parent OR the other anymore (they can also be made hybrids later-they don't have to be born that way). Pokemon can only truly be hybrids of two species (with an exception explained later on) but may show features of others that might be in their lineage. Hybrid pokemon will take one type from each species, but still have the same range of attacks each individual species would have. A grass/fire venusaur/charizard could still use poison type attacks, they just may not be as powerful as a pokemon with the poison typing.
Arceus is the only big pokemon truly missing in this setting. Legendary pokemon have simply hidden away, but are around. For the most part, there only exists one of each legendary, and some have changed their appearances to some degree to hide away better.
On the rare occasion a normal pokemon actively interacts with a legendary, that legendary may choose to bless that pokemon for whatever reason. Legendaries are not faultless and can be viewed similarly to the Greek Gods who have their own personalities, wants, things that impress them, etc. A pokemon looking to gain favor with Ho-Oh would probably have a different strategy than a pokemon looking for Dialga's blessing. These blessings offer otherwise regular pokemon additional power. They become stronger and tend to radiate a minor aura that reflects their newfound power. They may take on physical attributes similar to their blesser. Being blessed is the only way for a pokemon to have a third type, or a third hybrid species. A blessed pokemon cannot pass down its blessing in breeding, but may pass down the type/s.
Not every blessing is created equal, and if recieved incorrectly, can be seen as more of a curse. A bit of power bestowed with minimal care, or taking a powerful attack directly from a legendary can result in an unsuitable body recieving this curse of sorts. These pokemon will survive, but their less powerful bodies mean they may deteriorate over time in different ways. Often, this manifests in becoming simply too much for the pokemon to stand and they end up in a lot of pain and may become violent with no direction, similar to pokemon trapped in mystery dungeons.
Despite the fact mythical pokemon are more numerous than legendaries, sitting somewhere between standard and legendary, they've also largely tried to drop off the radar though they're easier to come across than the more powerful pokemon. These can be treated a little more like standard pokemon that can breed and have different varieties (except for maybe domestic). They're still held to somewhat of a higher standard, when legendary pokemon are largely missing and there's a strong desire to seek out something great and powerful to revere. The shaymin of the guild use this status to keep order in their land, though they are by no means all-powerful. There are arguments as to whether true mythical or standard-legendary-blessed pokemon are more powerful.
Everything about legendary blessings also apply here. Mythical blessed pokemon may still hold an aura of power, though nowhere near what a legendary blessed pokemon might. While a pokemon blessed by a mythical still cannot pass the mythical side down via breeding, mythical pokemon themselves can reproduce. A shaymin who has blessed their jolteon partner may have shaymin-eevee offspring together, but if that shaymin-blessed jolteon were to have kids with a standard vaporeon, they would only have eevees (though those eevees could have the additional grass typing).
With many having found a way to this universe during the events of the alola-based games, ultra beasts have naturalized at this point and may be treated as any other standard pokemon (aside from necrozma).
These oddities can come about in different ways. Some pulled through spacetime by whatever odd force makes mystery dungeons endless, others changed by the chaotic forces of the lands. An iron bundle could wake up one day in a mystery dungeon with no clue as to how it got there. An amoongus could find itself suddenly changed after an extra dangerous exploration as if something ancient in its dna woke up.
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