To my knowledge, gender is thought of as a complex of at least two aspects: those differences between what is thought of as masculine and feminine, and the roles assigned to genders. In these cases, gender pertains to the ideas of gender identity and further gender expression. In other cases, gender is also thought of as including or being inextricably related to biological sex.
Gender isn't "made up" in the same way that words aren't "made up"; it is an aspect of society, but unlike words, gender is with some relation to observable physical biological expression, usually termed as 'sex'. But the concept of gender as I have seen it used is certainly depending on society, rather than physical sex.
Variations within sex (such as intersex) is another question and is unrelated to gender, as far as I know.
The problem with "gender" as a concept is that they hijacked existing words ("man" and "woman") that were previously used for "sex". This can be clearly seen by trying to define these words without reference to sex - you can't except by defining them recursively ("men are people who identify as men"), in which case, I think a better choice would be to invent new words for this new concept.
There's biologically driven behavioral differences between male and female due to genetics. These differences are on a spectrum that varies between individuals but they are clearly visible in aggregate. A lot of recent political rhetoric bulldozes over the obvious differences between male and female to the detriment of our species. For example, the push to have the same physical standards for male and female in the army and elsewhere will inevitably drive the standards down. Males are mentally and physically tougher by design. And before anyone calls me sexist, they also die younger. Everything is a trade-off.
I believe that denying these differences is sexist, as it results in laws that affect males and females unequally. Should sentences for violent crimes be shorter for men because higher testosterone levels make them more biologically prone to violence? This is dangerous political territory bordering on eugenics but an interesting question nonetheless.
Gender isn't "made up" in the same way that words aren't "made up"; it is an aspect of society, but unlike words, gender is with some relation to observable physical biological expression, usually termed as 'sex'. But the concept of gender as I have seen it used is certainly depending on society, rather than physical sex.
Variations within sex (such as intersex) is another question and is unrelated to gender, as far as I know.