A khajiit's breed is determined by the lunar phases during their birth. Argonian breed is determined by how many times they lick the Hist tree.
Game - Khajiit breed
Arena - Ohmes (identical to Bosmer)
Daggerfall - Ohmes-raht (similar to Bosmer, except with body fur, cat ears, and tails)
Morrowind - Suthay-raht (bipedal cats with claws, digitigrade feet)
Oblivion - Suthay (identical to humans, except with fur, animal heads, and tails, no claws, plantigrade feet)
The Argonians in Arena may not be Argonians at all, but Kothringi tribesmen; it is widely rumored the Kothringi were hybrids of Argonians and Nords. Daggerfall and Oblivion, interestingly enough, used the same kind of Argonian: anthropoid lizards with plantigrade feet, and females with boobs. Morrowind-type represents the farthest difference from human anatomy.
There is no explanation for why all beast race members in a particular game are of the same breed.