I understand the point you are trying to make here, however, the is such a thing as gender. What is incorrect is to confuse linguistic gender with human gender. Gender is a thing that is used in plants, animals, primates, and sapiens, to describe the nature of their contribution to the reproductive cycle. Gender in Linguistics which is way more important in some languages and not so much in others, often reflects the physical world, but not wholly or completely and there are obvious variations. English does not itself make great use of linguistic gender, and this has been further confused by changes in the way we use language. For a long time, we could talk about mankind when referring to the whole species. These days that is likely to be misunderstood, so we would more normally these days spea of humankind. Like all things there are wins and losses (I preferred Psalm 8 when it rendered what is man that you should be mindful of him but now we are asked to read what are human beings that you are mindful of them mortals that you care for them? Of course, we further confuse things by using verbs as nouns and nouns as verbs.
Do you have evidence to back up the claim that "there's no such thing as gender" or is this some sort of grand pronouncement that you're hoping we'll just take your word for? But if you're right that gender doesn't exist, why do you suppose so many people get their undies in a twist when a person has the temerity to live openly as an alternate gender, gender fluid, or as no gender?
Here is a helpful article on the subject of gender (including gender identity) from Psychology Today. https://www.psychologytoday.com/us/basics/gender
These people are confusing merely having a personality with 'gender'. It's made up and was only previously ever used in linguistics. My sex is female. That's all there is to it, there are no other components. I may be a tomboy; that's a personality trait.
Another point that should be made is that this statement is scientifically incorrect. According to the Intersex Society of North America, approximately 1% of people cannot be so simply classified based on their biology. There are at least 16 biological and medical conditions that deviate from the usual male or female classifications. Examples include Klinefelter’s syndrome, adrenal hyperplasia, and gonadal dysgenesis. But...you know...why should science get in the way of a good biology argument.
So what is a gender, then? Why wasn't it recognised throughout all of history? Why is it only now? Gender is really being used because brains are sexed organs, and if your brain is differently sexed to the rest of your body you are transsexual. That's really it. Some people think this constitutes a 'gender identity', which is made up. It's an issue of biology, not psychology or 'gender'.
It's an issue of identity. The term "gender identity" has been used by psychologists since the sixties to address a very real phenomenon. Saying something isn't real because it isn't biological sounds awfully materialist for a religious chat forum. If your issue is that you prefer the term "transsexual" to "transgender" it seems like you just want to argue semantics. The dominant term in common parlance is "transgender" or gender nonconforming, and I thinks it's a better descriptor for the issue at hand.
So you agree that sex is chiefly an act and not an identity and that we are discussing distinct concepts.
Wherefore let him that thinketh he standeth take heed lest he fall. There hath no temptation taken you but such as is common to man: but God is faithful, who will not suffer you to be tempted above that ye are able; but will with the temptation also make a way to escape, that ye may be able to bear it.