I have doubts on this. She has feathered wings, like the cat. The dragon has normal medieval dragon wings. If anything, I'd say she's the same species as the cat.
Correct me if I'm wrong, but I don't remember Alec introducing himself to the dragon. The dragon's sole duty was to guard the kingdom. The details on why aren't clear, but I don't think it was directed at a specific enemy. So, what they know about Alec has little to do with the dragon.