Posting this for the lurkers/readers, not necessarily for the edification of OP, who I suspect has no interest in edification.

If we were to make a similar argument about other classifications, it would be: Backbone (the feature) = mammal (the category). Except that isn't true. Any number of vertebrates could have backbones, and so the presence of a backbone does not indicate one is a mammal.

You are intentionally confusing categories and making fake muddles where none exist.

No one other than you has ever suggested that "the presence of a backbone" indicates that an animal is a mammal! Mammals are a particular kind of vertebrate with certain features - warm-blooded, fur or hair - that not all vertebrates have.

A main distinguishing feature of mammals is that female mammals typically give birth to live young of the species (as opposed to, say, laying eggs that mature and hatch outside the female body), and female mammals have the potential capacity to make and secrete milk that is capable of nourishing, sustaining and providing immune benefits to newborns/the young.The clue is in the name: mammals comes directly from the Latin words meaning breast - mammalia, mammalis, mamma.

So then it seems "those that produce eggs are women" or "those that produce sperm are men" is incorrect, as one does not need to have/produce eggs to be a woman or have/produce sperm to be a man.

Again, you're intentionally making muddles where none exist, a) by confusing individuals with the categories to which individuals belong; and b) by taking information relevant only to some specific sexually-reproducing species, namely those few and rare ones where sequential hermaphroditism occurs, and pretending that this information is relevant to and descriptive of one particular sexually-reproducing species, namely Homo sapiens aka human beings.

No one other than you has made the the claim that "those that produce eggs are women" or "those that produce sperm are men"!

What has been said, again and again, is that in sexually-reproducing plant and animal species, organisms develop along two clearly different pathways in accordance with the way evolution has arranged these species to perpetuate themselves - and organisms be categorized into two different and binary classes accordingly:

Female denotes the category of organism that if development is normal will have the potential capacity to produce mature eggs at some point in their life cycle; male denotes the category of organism that if development is normal will have the potential capacity to produce sperm at some point in their life cycle.

The words woman and women mean adult human female; man and men mean adult human male - not "those that produce eggs" or "those that produce sperm."

Anyone who can't see the difference between the definitions, classifications and species OP (and posters like the misogynist crank vintologi) keep purposely mixing up is beyond helping. But then clarity and specificity are not the aim of these fellas, right?