Ooh! I've spent a good amount of time thinking about this. As to what the Bible says about it..... nothing, it would seem.
I see transgender as another form of intersex. Sure, "God created male and female", but intersex people--rare anomalies that exist because of the fall of man--exist. If we define gender by reproductive organs, then how ON EARTH do we dictate the gender of someone who was born with both a vagina and penis, or who has a mixture of both sets of internal and external organs? Usually doctors just pick one when the intersex person is a baby and does surgery to make their bodies look like the gender they assign.
So far as being transgender goes, science is still in the early stages of trying to make sense of it. There is some evidence to suggest that gender does have more to it than just our physical bodies.
For one, one scientific theory is that we have "body maps"--this theory has been used to explain stuff like sleep paralysis and phantom limb syndrome. I have talked a lot to trans people, and some of them have talked about feeling vaguely as if there should be a penis where there isn't one, or (for the male-bodied ones) feeling as if their penis shouldn't be there. Basically, it's not listed in their brain's body map, which creates a vague or unsettling feeling of it not belonging. Some non-trans people also exist who experience something similar, ie feeling as if one of their limbs doesn't belong even though it clearly exists.
And conservative Christians have been arguing for YEARS that "men and women are very different".
Now they want to say that it's only about genitalia??
This is also an interesting read. It's about a village in Africa where children grow up with female anatomy but then when puberty hits they become male.
I don't think it's in this particular article, but in an article on the same subject that I remember reading a few years back, what struck me was that one of the people interviewed said that when they grew up as a girl in their childhood, it always felt "off" somehow.
So far as how to handle gender dysphoria if you have it, I can see arguments from both sides as making sense in their own right and I'm not 100% sure what I would do in those shoes.
A loved one came out to me as transgender last year, and I know of nothing better to do than support them however I'm able. How they choose to handle their condition is not my place to say imo.