You know, if your beliefs actually have good arguments behind them, you normally don't have to make up stories.
Also, this isn't even how this would work. Doctors see in a person's history if they are transgender or have had a sex change operation. In a real situation a doctor would know if the person has a prostate or not. So not only is this story made up, it isn't even plausible.
People who are transgender have a disconnect between their brain and the rest of their physical body. There is actual science, including chemical and genetic testing, as well as brain scans, to support the notion that transgender people actually do have brains and hormones that match their identity. Nature is full of all kinds of strange things and problems happen in fetal development all the time. Transgender people tend to have brains and hormones that resemble those that we would expect to see in gender they identify as. http://sitn.hms.harvard.edu/flash/2016/gender-lines-science-transgender-identity/
This may be true but there are so many gender identities and pronouns now that it has become absurd. To keep things simple, I tend to refer to people by their assumed biological sex if it is obvious, not their gender identity which would have to be guessed at anyway.
Given that this is a medical condition, it raises all sorts of questions as the the appropriate treatment that haven't really been dealt with. Is this a condition we should try to test for, detect and treat in different ways at an earlier age, or is assuming a different gender identity and possibly having a sex change operation really the best or only route? How do non-binary gender identities fit into this explanation?
You are biologically a man or a woman based on your chromosomes. Identifying as something different doesn't change that. Don't get me wrong, you should be able to do whatever you want with your own body. I just think the Nazi like attitude that you have to refer to somebody by whatever made up gender they choose or you are a sexist, homophobic bigot has gotten a little out of hand in some cases.