I'm a little unclear on how the furry identity works. Is it like an LGBT+ thing where you just are this thing regardless of your feelings or desires, or does wanting to be a furry make you one? Like, I've fought against being trans much of my life, but now I see I pretty much always was. But I don't know if liking the puppygirl idea makes me a furry, or if that's something I have to have always been? (this is not a reaction to a recently popular puppygirl, I've meant to ask this for a while) I also may be terribly misunderstanding furries; that's a taboo subject where I live, so I don't know much.
As a non participant who has never gotten it, people are more than welcome to do things I don't get, but I've tried to get my head around it and I don't think there's really any specific criteria. If you call yourself a furry, you're a furry and what that means varies quite widely from person to person. Some kind of association with anthropomorphic aminals seems to be the only criteria and even then I'm not sure how strict that is.
FurScience has a few classifications (that they use mostly for survey purposes). Most people just appreciate and enjoy the art and culture, at the far end of the spectrum, people identify as part- or whole-animal, or that their past life was a specific animal. These people are in the minority, and I'm not saying it's right or wrong, just what is common and uncommon.