So OP she just needs a few basic tools now and she’ll join more of the dots up herself later. You are not ‘transphobic’ you are happy to see ALL gender non conforming women doing what they like. if some of the women think they are actually male or NB then that’s fine and up to them.
Your daughter needs to know that ‘Gender identity’ is an optional personal belief system separate from, but informed by ‘gender’ ie changing cultural stereotypes of ‘masculinity’ and ‘femininity’ imposed on both sexes for the purpose of male advantage.
All women gender non-conform in some ways because gender ie‘femininity’ is a virtually impossible standard for any woman to fulfil. This is because its so subjective, culturally shifting and internally radically inconsistent - the whole POINT of gender is it’s a hierarchical trap for women always running to keep up, in order to keep men on top, provided the men fulfil some basic obligations of ‘masculinity’.
So that’s why you agree strongly with your daughter with what she says about the bad old gender norms, that’s why absolutely loads of people of all ages and life experiences think gender is shit. In particular she needs to see that this includes you, one of the ‘old and ignorant’ women who have struggled against misogyny, in your life because of the restrictions and attacks misogyny it makes on you, will do on her, constantly does to all women.
Tell her it’s great that she cares about equality. It’s a great thing that we all need to care about! But the key thing is, equality is for EVERYONE or it’s not equality. That includes for her, you, all women, and every group however they are defined by others or wish to self describe around themselves in relation to any characteristic or belief they hold- not just around biological sex obviously. We are all entitled to have equality together.
And as your daughter has partially begun to see, we don’t have equality yet for very very many groups in society, including women, people of colour, people with disabilities, people who are same-sex attracted and many many more.
Sorting that mess out for the benefit of women, doesn’t include letting in men to women’s spaces and giving them women’s opportunities and doesn’t include not safeguarding girls and women. Because the really problem here has nothing to do with gender identity, it is with some MEN who are well documented to take advantage of any access to girls and women they can get.
However those men identify their gender identity (if they know or care about that stuff at all) is beside the point. Male pattern offending persists after gender identity transition but the key thing to know is in the first place that there IS such a thing as male-pattern offending.
You have sex-based solidarity with transmen and female ‘NB’ people as with any female person who in whatever way doesn’t perform gender to the shifting standards of the day, because we know society does in many ways punish them. Therefore safe zones of whatever kind just for women are essential.
If your daughter can grasp that then she’ll see for herself in time that ‘gender identity’ as it is currently being pushed at her, is a men’s sexual entitlement campaign built on some vacuous man-pleasing shifting sand. Importantly hopefully she will start to feel reassured and inspired that lifelong she herself can have the pleasure, inspiration (and if necessary refuge) of the company of other women.
She’ll come to see its OK to say it’s up to men to put their own fucking broken house in order, it’s not women’s job to do it for them. Men need to do the heavy lifting work for themselves around freeing themselves and their male peers from ‘masculinity’ gender norms. And crucially, that male violence isn’t women’s problem to solve. Making all women less safe doesn’t make some men more safe. That’s not how male violence works. The problem needs to be named and dealt with at source.