At the core, the reason why I'm not comfortable with viewing myself as a dragon in a non-roleplaying context is also the reason why religion and spirituality doesn't work for me. My identity is strongly based on my observations and understanding of reality, of myself and everything around me.
Both religion and therianthropy give people an explanation they otherwise wouldn't get from science. It's about the why rather than the how.
Religion explains why the universe and all of us exist.
Therianthropy explains why one feels drawn to the idea of being an animal or why they exhibit unusual behavior — because on some level, they are an animal. And yes, this often applies to psychological therianthropy as well.
The problem is that those *why*s usually contradict my understanding of reality, and I can't stand that. I find it much easier to accept that there is no god, given that there's no proof or any "proof" can be explained without a god. Likewise, I rather view myself as fully human because there's nothing to prove the contrary without a shred of doubt.