* There is no satisfying answer to this question because "identity" has no singular definition: sometimes it's something intrinsic; other times it's something that's completely acquired. People cherry pick and apply the analogies that conform to their preexisting beliefs or their political affiliations.
* It's not the case that a sex reassignment surgery is just a cosmetic alteration -- a person that receives massive doses of hormones is changing their internal biochemistry, the way their muscles grow, and even the way they think at a fundamental level. If it's inauthentic to say that a transwoman is a woman, it's also inauthentic to call this person a man. As an aside, a 'gender spectrum' may not be the worst form of classification in this case.
* Anti-trans folks talk about the genotype as though it's hardwired into the human soul. There is evidence now that gut bacteria release chemicals which alter how the body works, how the brain works, and how genes are expressed and activated via epigenetics. Gut bacteria aren't intrinsic to the human body, but they have a colossal effect on both phenotype and genotype. Even social interaction can activate or deactivate genes. Genes aren't a sacred codex, but are altered (if not structurally, then functionally) all the time via the most trivial causes.
* It's not known whether or not transsexuality is itself a genetic phenomenon. If we discover a "transsexuality genotype" which makes people feel like a different gender, then transsexuality would be natural enough to deserve the same status as classical gender differences. The same appeal to nature can be made, for transsexuality, as is often made for classifying gender according to XX/XY chromosomal differences.
@squark. For what it's worth, I'm still on the fence about whether or not this whole thing is just a temporary fad. I'm aware that people can be convinced to hold the most outrageous beliefs and commit the most self-destructive actions. I really, truly cannot fathom these people's state of mind. I'm sorry to say, however, that the arguments levied against transsexuality aren't particularly decisive.