There are quite a lot of definitions of "therianthrope." Most of them I dislike, or they don't suit me. Here's the one that works—the definition that I use, instinctively, out of habit.
I don't believe that a therian has an animal soul as such; or, more specifically, I think of a soul as what makes you you. Not a blob of spiritual energy, but a way to visualize the self. So a therian could definitely have an animal soul, but the soul would be a product of their therianthropy, not the other way around.
I don't believe that there is physical proof of therianthropy. Weres do not have DNA involving any sort of non-human material; otherwise the phenomenon would have been charted scientifically by now in some form or another. Nor do I believe that it is entirely based on faith, because there's certainly a lot of concrete evidence that points to therianthropy as being real.
That evidence, though, doesn't say to us "This person is really an [insert animal here.]" That evidence says merely "This person really does act like an animal, and probably really does think like one, too." You take that evidence, and you take your body dysphoria, and you can prove therianthropy. Not scientifically, certainly not, but you can prove it to yourself.
Established on what others have observed about my behavior socially and mentally, and the way I handle my body, I am more like a cat than a human in many instances. I define myself as a cat because that felinity is not just in the way I behave; it's also in the way I exist.
These qualities go deeper than mere resemblance, because I do not force them. They come unbidden. I do not tell myself I'll be territorial or skittish because that's what a leopard should do. I am territorial and skittish because that's what a leopard does do. Yeah, enough with the italics, I know.
I usually try not to theorize about why I'm like this. I figure that if if I act like/am a leopard, and I don't harm anyone, I might as well keep on doing it (and enjoying it.) People can think of me as just "acting like a cat," I can think of myself as "literally being a cat," things work out fairly well.
Therianthropy is a state of being in which the therianthrope exists, lives, thinks, has instincts, and often acts as a non-human animal. Not "like," but "as." For most people, the level of humanity or animality varies from day to day and situation to situation.
It's a lot deeper and richer than that, and is often more than can ever be described in words. Is therianthropy special—does it set you apart from the crowd? Not really; you still live a pretty basic human life. Is it always fascinating, and does it always offer you great insights? Definitely not. I'm just one of those people who likes thinking about what I am.
Which is why, despite its often being mundane and unproven and strange and uncomfortable, I like writing about therianthropy.