They don't think they are these things physically. As they describe it, it's more like they have the souls of these creatures. They are dragons in human bodies in their minds, much as a transsexual is a man in a woman's body.
I don't think there's any similarity between Otherkin and transgender experience. "Man in a woman's body / woman in a man's body" is a terrible description.
Transgender people are fully aware of what their sex assigned at birth is, their chromosomes, and sexual organs. The difference is how a cisgender and transgender experience gender.
People perceive their gender, its something the brain does. Its analogous to the way ordinary, healthy people perceive the position of their limbs, the boundaries of their body, etc; most of the time, you aren't actively aware that you even have these perceptions until something disrupts it (brain injury, psychedelic drugs, optical or sensory illusion).
Gender is very interesting from the biological point of view. Barack Obama only perceives himself as a man because his brain is creating his gender perception. If Barack Obama were born with complete androgen insensitivity syndrome (CAIS), his body might produce testosterone, but he could never use it; whatever parts of his brain which contribute to his perception of gender would never masculinize, nor would any other part of his body, and hypothetical Barack Obama would look, sound, behave, and perceive his gender as a woman without ever knowing anything was out of the ordinary. At least until his parents worry why their teenage daughter hasn't started her menses, and a trip to OB/GYN reveals the presence of undeveloped male organs, absence of a womb, etc. Really fascinating stuff. Experience and perception of gender is something the brain does, not the chromosomes, and not the sexual organs.
In most circumstances, cisgender people percieve their gender without ever being consciously aware of it -- until something disrupts it. For example, a cisman may become intensely, consciously aware of his gender if he were made to shave his legs, wear heels, and strut his stuff in public. A cisman in that position might feel
very uncomfortable in his own skin at the moment.
Transgender people experience their gender just like everyone else, they just don't have a perception which aligns with the experience of cisgender people. The mismatch between a tg person's experienced gender, and the gender others treat them as having, is mentally jarring and a source of wordless feelings of discomfort. The feeling is multiplied by shame and guilt if others in their immediate surroundings mistreat gender-variant people with disgust. Most pre-transitioned people squash these feelings in a mental box as a defense mechanism so they can function in their community.
Post-transitioned people are very much aware that they are transgender, but aren't usually bothered by that knowledge. Most successfully transitioned people don't suffer discomfort after they've integrated into their target gender.
That roughly captures the most common first-person experiences of a transgender person. I don't know whether this is even comparable to the first-person experience of someone who spiritually identifies with Ice Dragons or otherkin.
