Despite their diversity in shapes and forms, the majority of Pokemon species seem to have seperate sexes, and reproduce via sexual reproduction.
Despite having a very expansive genome and such radically different morphologies between eachother, Pokemon as a whole only have the amount of genetic diversity between eachother as just a single genus of animals. The genome of an Onyx compared to a Weepinbell is as different as that between a husky and a coyote. A “species” of Pokemon is not determined by its nuclear DNA, but rather by specialized organelles outside the nucleus. These organelles seem to heavily regulate transcription of genetic information, and turn “off” large sections of the genome and control which genes are expressed.
Like how Mew contains “the genetic code of all pokemon”, so do all regular Pokemon too. The difference between a Pikachu and a Meowth isn’t nessesarily the differences between their genomes, but rather which parts of their genome do they express.
The organelles responsible for these differences in gene expression are also strictly inherited from the mother, in her egg cells (much like how we inherited mitochondria from our mother). This means while a Pokemon who’s mother is a Meowth and father is a Pikachu does have half the genes of both it’s parents, it will only inherit the organelles from it’s mother. Hence it’s “species” will be that of it’s mother, and will hatch as a Meowth regardless of it’s father.
So yes, using artificial insemination you can breed a Skitty and a Wailord together, but the babies will always come out as the mother’s species as opposed to some sort of hybrid that would kill a female skitty to produce.