So this is where your ignorance lies. At least it had a reason...surprise surprise.
I'll give you three phrases regarding a trans person.
"Mr. Jack's sex at birth was that of a woman. His gender was always that of a man though."
This phrase is completely clear and correct under current meanings of the words.
"Mr. Jack's sex at birth was that of a woman. His sex was always that of a man though."
This phrase is partially correct. Second instance of "sex" is used incorrectly however. Then there's the fact of it being used twice in a row saying two opposite things. It's not relevant here because the second use was incorrect to begin with.
"Mr. Jack's gender at birth was that of a woman. His gender was always that of a man though."
Here we have an error as well...but not in use of words. The word gender is used correctly both times. The error is more of a matter of style. Repeating the use of the same word in consecutive sentences (or within one sentence) was always a problem that is to be avoided. Here we have also another problem in that the word is used in two related, but separate contexts...which are opposite of each other. That makes this phrase correct on the meaning...but something you'd get an F for at school.
But we can fix that very easily.
"Mr. Jack's gender at birth was that of a woman. His gender mentally was always that of a man though."
And voila. Problem solved, except that it still looks so-so with the word "gender" repeated in consecutive phrases. However since the word "mentally" was added, the issue of misconception is solved because now not only first sentence have context (at birth), but the second one have as well (mentally). Something the phrases above lacked.
Transgender people are proof of there being a possibility of someone having two genders that are different from one another (mental and biological)...or of there being a person whose sex and gender are not the same. Both of these statements are correct. They merely state the same thing in two different ways. You know...the whole purpose of synonyms existing.
I think it's less a matter of it being "polite", more a matter of it having a second meaning that is considered taboo. If public use of the other meaning of that word did not risk fine for obscenity, sharp looks and possibly being raped...it would be used a lot more often even outside scientific and medical circles.