I don't know if you want to define the species based on
phenetics alone; right now the best we can try to achieve without a detailed understanding of physiology would be a
folk taxonomy with educated guesswork thrown in.
From an evolutionary standpoint, I would imagine that dragons might be split across the Animalia kingdom, based on the varying number of appendages. For example, your classic wyvern has four appendages (two wings, two legs), Oriental dragons have 4 legs, and I expect lindwyrms, the Oriental dragons with two legs, and those feathered ones with only wings have vestigial limbs to bring the count up, like
snakes. Most of these guys could be placed as a group buried somewhere in Reptilia; easy peasy.
European dragons are the problematic ones. The 4-limb rule of thumb is older than Mamalia; it's though to date back all the way to when we were fish, and has never changed since. This leads to three possibilities:
* European Dragons evolved on a completely separate evolutionary path than every other land animal in the world, a path that produced ONLY the European Dragon. (Doubtful)
* European Dragons are NOT a separate species, but a mutation or
birth defect of the wyvern that produces a extra pair of limbs, probably stunted and non-functional.
* There are no European Dragons, just wyverns. Early historians got their information from witnesses who were either panicked or stupid, leading to a mistaken impression that the monster that just ate Bessie had more limbs than it actually did. (Most likely)