I'd say as far as race goes, this is a question of a few individuals - this isn't exactly a thing. So I'll limit myself to the gender part.

I guess I would say because we generally view people's inner self as having primacy over the outer self, and hence respect their right to try and make their exterior reflect their inner self? Mind over matter, so to speak - seems like a concept Christians could generally get behind.

It IS in fact defined as a mental health issue (now that you clarified how you defined this), in the sense that gender dysphoria is indeed included in standard lists of psychological disorders, like the DSM 5. It's just that the best solution for it in many cases is to accept it rather than fight it, because after all why not do what makes you happy, if it doesn't hurt either others or yourself?

In that sense it's different from the large majority of other 'mental health issues' which either make the patient unhappy in a way that cannot be resolved, or cause danger or harm or impairment to the patient, or pose a risk to others (or some combination of those).

I was just pointing out how vague a description 'mental health issue' is - it could mean almost anything, down to someone waking up in a bad mood for no discernible reason. But you clarified you meant it in a much more specific sense, so okay.