Stag is the name of the male red deer; females are called hinds. With white-tailed deer and mule deer, males are bucks and females are does. Thus Lily’s doe does not actually match James’s stag, a fact that I'm sure Severus was delighted to point out. Smirk!
I never knew that. I always thought that hind was just an old expression for a doe that wasn't used that much anymore. I also thought that stag and buck were interchangeable names for a male deer. See, you learn something new from reading comments!