Man of Honour
I think she is irrational but despite that we actually have a lot in common and have a lot of laughs and shared hobbies so it's a storm in a tea cup tbh.
I am tolerant of people with different opinions to mine, no matter how silly I think they are. It's important to mix with people you don't agree with even more so these days in my opinion because we live in a world where lots of people exist in echo chambers and spout rubbish completely unchallenged.
I think that gives me the moral high ground and I'm quite proud of it to be honest. It doesn't mean I'm right but at least I make an effort to understand the opposite position.
I am tolerant of people with different opinions to mine, but I am not friends with people who are irrationally prejudiced to such an extent that they choose their irrational prejudices as the way they identify themselves. Even more so when their irrational prejudices target me personally and they explicitly target me personally.
I think some degree of equality is necessary for friendship. She regards you as inherently inferior. She thinks you deserve inferior treatment in all things - legal rights, employment, education, everything - solely because she's sexist and racist. That's not how friendship works. Of course she thinks the same of the billions of people she thinks are untermensch, but when it comes to the relationship between you her targetting of you personally is most relevant.
I have known some feminists and other irrationally prejudiced people who are otherwise decent people. I'd mix with them. I'd listen to them. I'd be civil to them. But I wouldn't be friends with them, especially if I was their target. No matter how fashionable their irrational prejudices are.
It's not like one person supporting one football team and another person supporting another football team. It's not even like different people supporting different political parties. It's a far more serious division than that, all the more so when one person targets the other but still very serious when they don't. For example, I know someone who is a whitist. They think "whites" deserve superior social status, legal rights, etc. They think "blacks" in particular are the cause of most problems. Yadda yadda. Just like your "friend". Same way of thinking, slightly different targetting. I'm in their favoured biological group so I'm not personally targetted by their views, but I still couldn't be friends with them. I can tolerate their racism on the basis of free speech, but it's repulsive enough to me to make it impossible for me to be their friend even if I wanted to be.
As an aside, I'm curious as to why she'd want to be "friends" with someone who she regards as being so inferior because of their sex and "race" that she thinks they're not even worthy of being allowed to have an opinion. That's a rather strange position to take. I can understand irrationally prejuduced people, even extremely irrationally prejudiced people, wanted to be friends with one of their targets because they don't target them personally. There's ample evidence that humans have a small group of people (probably ~150) who they instinctively understand as being people and for everyone else they only intellectually understand them as being people. It's therefore possible for an irrationally prejudiced person to have one or a few people in their target group(s) inside the small group of people they instinctively understand as people and therefore genuinely not think of them as their target group. But that's not the case here - she explicitly targets you personally, so she does include you in her target group.
You know she's targetting you. You know she thinks that you personally (in addition to billions of other people) are unworthy of equal treatment and always will be. You should be wary of any apparent friendship. How can she really be your friend when she regards you as so inherently inferior that you're not even worthy of having an opinion? When she actively promotes prejudice and discrimination against you? When she regards those things as so important that she defines herself with them?