Where is your evidence that trans people don't have the brain of their opposite sex in them?

Where is your evidence that persons with male bodies and XY chromosomes in every cell have brains that are female with XX chromosomes? In utero, brains don't develop separately from the rest of human bodies.

If gender identity is not a real thing, why do people who think they are not the sex they were assigned at birth and were born in the wrong sex and body exist? What's going on in their brains that makes them think they are in the right or wrong sex and body?

Gender identity obviously seems real to people who claim they have one, but it's not in any way observable or evident to others, nor is it verifiable or measurable by any objective means.

Gender identity is like souls; the claim that some people have them is matter of emotion-based faith, not of empirical fact.

As to what's going on in these people's brains that makes them have these particular delusions, sexual fantasies and obsessions, it's not clear. Also, not the job of GC to "explain." Coz we're not the ones making these fatuous claims.

Trans, and nonbinary people use male and female to refer to gender identity, not sex. Why should "male" and "female" not be defined to mean one's gender identity aside from one's sex?

Bully for them. But niche special interest groups don't have the right to redefine by fiat and diktat commonly-understood words used by the entire human race.

Female and male are biologically-defined terms that refer to the two distinct types of plants and animals in sexually reproducing species - one of which has evolved to develop the potential capacity to produce small gametes, sperm, at some point in time; the other of which has evolved to develop the potential capacity to produce large gametes, eggs or ova, at some point in time.

Most humans, and all of the millions of other sexually-reproducing plant and animal species, do not have "gender identities." What's more, humans and other animals over time have evolved to be very adept at distinguishing the sex of plants and animals of other species as well as sex within their own species.

Dogs can tell the difference between men and women, as can infant humans. Many crops (apples, almonds, broccoli, etc) depend on pollination by bees - basically assisted fertilization; in order for this to happen, bees have to know the difference between the male and female fruits and vegetables involved. Asking animals and people to use male and female to refer to "gender identity" rather than sex means going against the entire history of evolution and expecting the human and animal kingdom to deny what their senses tell them.

Are there any disadvantages that come with male and female being used to refer to other things besides sex?

Yes. Agriculture would completely collapse if neither humans nor other species could identify the sex of plants and animals - and had to rely instead on plants and animals somehow revealing their inner "gender identities." Pollination of plants and the mating and reproduction of agricultural animals would become much more "iffy" and eventually would stop.

Imagine if farmers couldn't distinguish between bulls and cows based on their clear biological differences, but instead had to suss out the animals' gender identities. Consumers would end up with bull semen in their coffee, on their cereal and fed to their children.