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.
Humans have identified the self in animals for as long as archaeological history records. The very earliest surviving figurative sculptures and illustrations of human beings feature animal features. It seems to emerge naturally from human perception, the inclination to identify with particular animals and to produce images of animal like humans, and human like animals. How this has been culturally contextualized has varied tremendously throughout history. The furry subculture is simply a modern manifestation of this impulse to produce and identify with such images and representations.
I'd speculate it's strong current within the LGBT communities is a side effect of contemporary western culture's lack of an established cultural context in which to exercise this artistic and conceptual impulse, naturally leading those who feel it strongly to be or to sympathize with people who are not well-represented within dominant culture in other ways.