Taking place in the Mystery Dungeon universe. Hybrid pokemon are relatively new, as far as the timeline in the world history is concerned. They have existed for the past couple hundred years, long enough to where it's pretty commonplace to see them in society.
Hybrid pokemon came about as a gift to the world from Xerneas himself- being a deity of life, with the blessing of Arceus. He had fallen in love with a Girafarig lady, and longed for his own child. Thus, effectively getting rid of egg groups altogether and making it possible for hybrid pokemon to come about. Duet is the first born, and only child of Xerneas, and the first hybrid pokemon to come into this world.
Not every child will be a hybrid- a pair have a 50/50 chance in producing a hybrid child, or a "purebred" pokemon of one of the parents' species. Even a pairing of one or more hybrid parents still have an entirely random chance if the offspring is a hybrid or a purebred of one of their species. Hybrids will nearly always be made up from two pokemon species- and in incredibly rare circumstances a third species.
Where shiny pokemon are concerned, there are two different types. Common "normal" shiny pokemon are your standard color varient, they're honestly more uncommon than anything else, and not as rare as people might believe. Pokemon have roughly about a 1 in 4,000 chance of being born shiny- which, while may seem like a lot, but even when compared human world cities- this would be on average about 25% of the total population in an average city (as cities are generally around 100,000 in population or higher.)
Alternate shinies are considered rarer forms of pokemon, characters like butterfree's mate in the anime, or the purple kecleon brother in PMD games. They're still pretty rare, but still not a total surprise to meet pokemon with this coloration either.
Borrowing from the TCG, delta species pokemon are different types than they normally would be. However, their appearance is changed slightly as well to accommodate this mutation. Similar to region specific forms of pokemon, however delta pokemon can come about at any stage during the pokemon's life. As pokemon are adaptive by nature, this mutation occurs over prolonged periods of exposure to harsh environments that a particularly species is not normally suited for; though this change is a lot easier to achieve in younger pokemon. Some will purposefully seek out places that have been known to cause these mutations in their particular species / typing.
Take for example a pokemon like bellsprout, if it's introduced to a harsh environment such as a volcano for extended periods of time, they might adapt into a typing that can withstand that kind of location that they normally wouldn't be able to handle quite as well. They could harden their bodies, becoming a rock type and a fake or pseudo plant just like bonsly/sudowoodoo, as an example.