I think this question isn't able to actually
be answered, because being Furry means something different to every single one of us.
At least, none of the answers will be the "right" one. There is no right answer, in my opinion. Here's my take:
To be furry is to share an interest in anthropomorphic
(levels 1-4) animals with a multitude of people across the world. Whether for characters, desires to be a creature you aren't, a pagan or other belief that you actually are an animal spirit, (dare I mention it) exotic porn, or just for the hell of it, we all share the one trait of having an interest in animals; whether it's being one or just having characters who are one.
I know a few Furries-in-spirit who have no 'Sona, by choice or by indecision, and those without the fursuits that gained us our notoriety, but still share the same love for animals and anthropomorphs that the rest of us do. That is what being Furry is at its very core. We don't need the 'Sonas and fursuits to be Furry. Those just help represent ourselves. It's something that we choose to be, based on something we love.
I specify animals specifically because Anthropomorphism can branch to such things as household appliances (
The Brave Little Toaster, anyone?), furnishings (Disney's
Aladdin's Carpet), machines (Pixar's
Cars), and other inanimate objects (AI robots and automatons excluded). That's a whole 'nother ballpark.
I basically just said what Khleo did, but I made it longer. xD