One of my fave quotes (that never seems to amuse anyone else) is Victor Hugo (you know, the guy who wrote the book Les Miserables-- yes it was a book before it was a musical!):
Personally, what I think lies at the root of the fandom is a fascination. And at the root of that fascination are two instincts (fairly well documented by evolutionary psych guys) we evolved to have:
One is a fascination with living things, their forms, habits, and essential "character" or "essence" or "spirit." Thus, catness, dogness, horseness, etc, especially if you've ever been aroud cats, dogs, horses, respectively.
The other instinct is a fascination with/awareness of other intelligent beings-- and the reflexive attempt to try and understand or "model" what's going on in their minds.
When you put the two together, you inevitably have an inborn human fascination with the idea of animals with a human point of view (and by extension, intelligence, language, even hands and the rest.)
From there, I think it's fairly normal to try and elaborate this sense of fascination and awe, into a distinguishing part of one's identity. Like you said, people notice that a person has certain "catlike" features, ergo, they identify with a cat.
Through history, human spiritual practices have always found some set of inspirational ideas to latch onto. In some eras, it's the cycle of the seasons and vegetation growth, harvest, death, rebirth-- and the slain-and-ressurected king that goes along with all that. In others, it's the complex but regular patterns of the stars and planets. In others, the perfection and nobility of human form and reason. In many of the first, as with our ancestors over the millenia of the great hunt across the plains of Eurasia-- and I would argue, strongly emergent in the furry fandom-- the fascination is with animal forms and powers.
I agree that it's flimsy to go with some definition of "furry" most of us know misses the mark, like "anthing goes!!!"
On the other hand, I disagree that an especially intense identification with an animal is the most basic thing. The most basic thing is the fascination itself.
I think another thing that comes from identifying with an animal or species, is the whole make-believe game of "what would it be like to be such an animal," and the biggest deal for whitebread joe like me, at least, when it comes to that, is that animals obviously live largely free of the ordinary social baggage and jarringly artificial circumstances we humans find ourselves trapped in.
One of the words I hear furries use to describe this imagined state of mind is "innocence." However you describe it, this similar-but-different, social-but-not-trapped, uncomplicated, baggage- and disgust- and superstition-free attitude toward life that we can imagine having as animals-with-a-point-of-view, is the other part of what I've always seen as essentially furry, and I think you may have been getting at it too, in a way.
Anyway, that's my take on it. Inborn fascination with the idea of thinking animals plus the daydream of naturalness/innocence. That's furry.