What is this social category?

"Woman".

What is its purpose?

It doesn't really have one anymore, it just sort of happened.

How did it come about?

I'm not really sure. Sociology doesn't go into "origins of X" as much as anthropology. But here's a decent answer that seems okay:

"Gender was an ancient technology, the development of which was promoted by a Convergent-Evolution-like process; it happened independently in many different Human Cultures, and also spread from culture to culture due to the advantages it brought.

In a nutshell: The number of living adult Females in a band or tribe determines the maximum number of babies that can be born to it that year. Cultures without Gender Roles restricting Females from dangerous work lost more Females to illness and injury, and thus reduced their group's Reproductive Capacity. This, in turn, reduces the group's ability to recover from disaster, injury, illness, and other loss of life. Slower population growth meant those bands and tribes were generally out-competed by groups with more members. So too, culture is a tool [not] unlike evolution, and effective tools tend to inspire the creation or spread of similar tools in neighboring cultures. These are some of the reasons Gender Roles are so ubiquitous.

The end product is that, deep within ancient laws, social contracts, and what's accepted as normal for most cultures in the world, we find these Gender Roles buried. They're so ancient, and so deeply-tied to the rituals and jobs and ceremonies and other fundamental parts of a culture that they seem Biological or Fundamental. This, despite said Gender Roles varying widely from culture to culture. And also despite being rendered completely irrelevant in an era when mortality and maximizing reproductive capacity isn't a concern (and is even arguably harmful, given the survival risks posed by exponential Human overpopulation).

Side note: The invention of Gender Roles was likely an influence on our biological evolution as well; mild Sexual [Dimorphism] tends to emerge among species like Humans, which have been social for a long time. The non-child-bearers are specialized for certain tasks, or in ways that make their activities less risky. These traits or capabilities are generally not be as pronounced in caretakers / child-bearers not involved in these activities, to conserve bodily resources and energy that would be spent developing and maintaining those capabilities. A good example is the way elevated Testosterone Levels in Men and Women encourage higher baseline Muscle Mass, but require more calories to maintain ("Sex Hormones" do not have this effect in all species; solitary hunters and non-social animals tend to create Males and Females whose capabilities are a lot more similar). Eusocial animals like Mole Rats or many species of hive-dwelling insects with single Breeders in their communities get even more extreme versions of this; most have at least 3 genetic or "sex-like" morphs that create specific genders/"social castes", and some have twice that or more. But we're getting off on a huge tangent at this point."

Gender 'culturally evolved' to distinguish the sperm-people and the womb-people, and gender roles 'evolved' to tinker with what each people did. Like many traits, gender is no longer necessary for societies, but it remains as a vestigial 'organ'.

This does not mean that gender is good! I want to abolish it, myself. But it is important to understand your enemy.

Why are females placed in that category?

Because it 'evolved' to target womb-people.

What is the definition of that category, or the parameters of what is required to fit within that category?

If you go by "what it was intended for," then womb-people. But that would also exclude post-menopausal females, sterile females, etc.

But wait a second. This is easy. We can look at how society categorizes people, and use that to determine "who fits in 'woman." And yep, it's your phenotype, because that determines which category society places you in. Descriptivism over prescriptivism.

(Not quite satisfied with that one, but it'll do for now. And you'll note that it doesn't include trans women, while managing to squeeze CAIS inside.)

It is still circular and paradoxical for trans natal males to say that adult natal females are not the model of "woman".

Oh, yes, females are the phenotype that "woman" was constructed around. But its original 'evolutionary purpose' is irrelevant, what matters is how today's society classifies people.