Draenor and Azeroth seem to have multiple species in common, so either there was a lot of crossover and invasive species when the Dark Portal was opened or cosmic nature just isn’t all that imaginative.
Vrykul, Humans (and worgen), Dwarves, Gnomes, Earthen, Troggs, Mogu, and Tol’vir are descended from Titan Constructs. While a Azeroth is a Titan, they are not from Azeroth. Their constructs would be alien.
Silithid, trolls, elves, tauren, pandarens, jinyu, hozen are native. Elves are mutated trolls exposed to the blood of Azeroth. High/Blood/Void elves are further mutated by moving to the Eastern Kingdoms and mana withdrawal. Naga, Harpies and Satyr are mutated night elves. Goblins are native but they are the result of experiments from a Titan Keeper. Tuskar, kobolds, gnolls, murlocs, dryads, and centaur are probably native with their destinies warped by old god wars and the well of eternity (azeroth’s blood).
When the Horde brought the AU Mag’har into Azeroth, the planet also gained Goren, Saberon, and Botani. The Goren were a mix of tamed and wild Goren, with the wild ones running into southern Durotar. The Saberon headed off in the direction of the Northern Barrens, and the Botani headed northwest in the direction of Ashenvale/Azshara.