I think there are relatively good points to make on both parts of the argument in the case of pokemon, because the trainers often treat the species name as if it was a proper noun, and names are often shared. It works well enough for the purposes of the show, as the same species is rarely ever encountered twice in one episode.
However, I think much of that is motivated by the franchise's vested interest in forcing you to memorize the species names and in the real world, most people name their pets, and Pokemon enthusiasts resort to using the possessive case to distinguish between important…