I would say it is, regardless of it's acceptability. It's not something I have a problem with, or condemn but then nor do I automatically place a negative slant on the term mental illness.
It does however fall into a very different realm, liking the same sex isn't the same thing as believing you are a different sex, it's much 'milder' imo.
I suspect I have a significantly more objective view of these things than the modern world would deem acceptable in most cases.
The key distinguisher with mental illness is whether it causes suffering or disability to the person who has it. Now with most gay people, suffering or hindrance is not inherent but inflicted by society around them. So therefore according to most definitions it would not be a mental illness. Just a condition. If you want to argue it does meet those criteria because, e.g., it reduces likelihood of procreation, then you could argue on those grounds. But it's a weak argument because such is easily circumvented just by closing ones eyes and thinking of, well, probably not England.
But anyway, that covers it as far as "mental illness", imo. Gender identity orders have a greater basis for being classed as such because they reflect (I think) a form of body dysmophia. Like anorexia for instance, the person's perception of their body is at odds with the reality.
But even there it is open to debate. How much of the suffering of such a person is due to inherent consequences of the condition, and how much from social and gender norms they feel subjected to.
I would say that it IS a mental illness in this case, but it's not something I would argue it can be classed as such without doubt. If in a hundred years times its seen as normal (though not commonplace), "sufferers" might feel fine with it. Though I suspect not.
In either case, whether it is or it isn't, it's not something that is made better by exclusion or argument. Lacking the ability to re-write the brain's wiring (and I think that would be the less preferable option anyway, for most), sex-change operations are probably the only medical solution.
That said, I'm of the opinion that only a portion of such people are neurologically that way. I think there are quite a few who class themselves that way for psychological reasons and these people probably need help.
All the above just my opinion.