Good question OP! My answer is no, regardless of whether woman is a gender or not.
Let's start with linguistics -
Woman is a noun used as a sex designator for adult individuals of the female sex in humans.
Vixen is a noun used as a sex designator for adult individuals of the female sex in foxes.
Empress is a noun used as a sex designator for adult individuals of the female sex in chimpanzees.
Angeline is a noun used as a sex designator for adult individuals of the female sex in wolverines.
And so on. Here the noun in question has no other function than to categorise an adult individual of its species as belonging to the female sex.
There are certain connotations closely connected with biology - reproduction, growth, size and behaviour common to that sex (both mating behaviour and otherwise). So this word tells us something about the sex class but not everything about the individual female if we know something about its species.
So, as woman is a word for the female sex in this sentence, the answer is no it doesn't match because it isn't a gender, even though the word does tell us something more about that individual than just her membership of the female sex. But it cannot tell us much beyond biology and what arises from it.
Now let's take the doctrine of gender identity -
Woman is a noun that describes everyone who feels like a woman.
Well, that won't do because it's a circular definition. Let's try this again:
Woman is a collective noun for the norms, expectations and stereotypes prescribed to and proscribed for the female members of any given society.
These norms, expectations and stereotypes change frequently across both time and space. What this means is that something a woman must or must not do in one society can be precisely the opposite of what she must or must not do in another society. I grew up in a society where all women were expected to work full time and be politically engaged. At the same time elsewhere on this planet women were barred from work and politics.
But let's ignore the changeable nature of gender as a socially constructed mechanism of categorising males and females. Let's assume it's fixed and consists of a whole range of pairs of behaviours and roles that divide into those imposed only on females and those imposed only on males.
For the gender of woman to actually match the female sex, in my view we would need to fulfill three conditions: 1) these behaviours and roles should be a natural consequence of being female. 2) Women as a class should conform to them. And 3) the vast majority should want to do so. Happily.
But the history of centuries of women first secretly rebelling and later openly fighting to throw off the shackles of "gender" already means we cannot meet 2 or 3. But if this mechanism of categorising male and female people was arising from within and not externally according to arbitrary notions of proper womaning, we would not be fighting against it. So condition 1 is not met either because of our ability to refuse to conform to any or all aspects of that mechanism.
By my reasoning then, woman as a gender is forced upon the female sex and not a natural consequence of it. Something that is forced cannot be thought of as matching. So, it's a no, again.
Let's go with Judith Butler and say gender is what we do (what we have learned to do) not who we are (what we were born to do).
By that reasoning, woman as a gender means we internalise gendered norms, expectations and stereotypes associated with the female sex in our society and we perform accordingly. It's not about whether we choose to do this or whether we're happy about it. We just do it, because we were socialised into it.
So does the Butler version of woman as a gender match the female sex? Not in my view.
- Not all of us are socialised into the same behaviours and roles. And it isn't that hard to socialise children differently. (If it were, the norms, expectations and stereotypes would be fixed across both time and space).
- Many of us can and have learned to unpick our socialisation as adults, and to "perform" differently.
- For some of us the socialisation doesn't take at all. It chafes too much. It doesn't sit well with how we want to behave and what we want to do.
So, for reasons 1, 2 and 3 we've got a whole range of women from different backgrounds, upbringings and abilities not performing our society's version of woman as a gender. We don't do womaning right. But what is the word for us if it isn't women?
How can a category that an unknown quantity of the female sex does not fit into be thought of as matching the female sex? And what do those non-conforming females match with?
According to the doctrine of gender identity, if we as female individuals don't match the "woman gender" we should be classified as trans. Especially since gender dysphoria (being unhappy in the body we inhabit) is no longer a necessary condition for being trans.
But sorting us into the trans category without us having this as part of our personality (aka identity) would make a mockery of the concept of everyone being allowed to decide for themselves.
So, it's another no.