It's always seemed so strange for Darkrai to be a mythical Pokémon while its counterpart is just a regular ol' legendary. Maybe it's just me, but it feels so uncanny.
Partially. While things such as PID, gender, ability, and sometimes IVs* are randomly generated, all of these Pokémon have a preset nature, moveset, items, and most times IVs*.
*IVs are a particular case, since, as far as I remember, non-Super Battle Subway and Maison use Pokémon with randomly generated IVs, whereas "Super" modes have all set to 31.
Adding to what wrathsoffire76 said, Smogon keeps a full list of these preset characteristics (be it for rent Pokémon or for NPC ones), not only for Battle Factory but for every battle facility. Serebii and Bulbapedia have these lists as well, and while the latter ones are easier to navigate, Smogon is usually a bit more technical.