People who are into dragons are furries, it's not a literal term. Their furry subgroup is usually called scalies, which besides dragons and snakes include people who are into things like salamanders and other amphibians.
I'm under the impression that such taxonomies are less important to these in-groups than whether you're just into the aesthetic or get off on it sexually as well.
It certainly didn't start with 80s cartoons - anthropomorphized depictions of animals feature among the oldest written works. Aesop's Fables are 2500 years old, and the geographic dispersal of similar stories indicates that they originate even further back than that.
I'm under the impression that such taxonomies are less important to these in-groups than whether you're just into the aesthetic or get off on it sexually as well.