Going back a bit, I'm going to have to disagree that feminists should support males who claim to be women because they are victims of the patriarchy just like us. Firstly, I don't think many of them are victims of the patriarchy - as others have said, being able to redefine the category of woman to include male people if they want to is an expression of male power. Males claiming to be women were not leading lights in the battle for women's liberation/equality, depending on what sort of feminist you are. They often have very traditional gendered ideas about women, not to mention an obsession with porn. So they didn't and don't fight for us, but we should fight for them?
Also, of course women campaigned for other women to have the vote, to have women-only shelters, equal pay and to be equal members of society (a bit of a way to go there!). But this was all about removing artificial barriers for women. There was a point when women were not allowed to vote, or make a complaint of rape if they were married to the rapist, or sue their employer if they were sacked for becoming pregnant or even just getting married. What are the barriers set up by society to restrict the life choices and careers of men who claim to be women? None, I would suggest. Or at least, none not faced by men in general.
They are protected by the Equality Act. There has never been a time when they were not permitted to vote or own property because of having an inner feeling about what sex they were. The GRA recognises them as legally female on the back of a very straightforward process. (I think it should be repealed, by the way.) They benefit from all the advantages of being male. (How many women who claim a trans identity are voted Man of the Year, or Best Actor, or Businessman of the Year? How many head up male penile cancer or Shed groups? And yet, despite the relatively small numbers of men claiming to be women, they are routinely catapulted to the top of women's organisations, awarded prizes and given opportunities intended for women.)
The only hurdles they face are the moral and legal bars on invading women's spaces, or being accepted as lesbians. Why should feminists help them overcome those hurdles? How could it be done without harming women? I don't think it could be, more to the point I don't think those barriers should be broken down. Those demands go well beyond ordinary fairness or equal opportunities.
Regarding the patriarchy, I always understood it to mean a society which is based on the wants and needs of adult men, with very little regard for women and girls and to some extent boys. Talking worldwide, here. It is enforced in the first instance by social rules and in the last resort by male violence - this is the importance of sex differences. Wealth and class play into this of course, and in most but not all cases, sexual orientation (think of the Greeks - homosexuality would have been a meaningless idea to them since free born adult males slept with whoever they wanted to for pleasure and with women for heirs - and pleasure). Wealthy white males are at the top of the status tree in the Anglosphere and Europe, I would guess, and the population of the global north is higher status than that of the global south. But this is all a long-winded way of saying that I see patriarchy as about the primacy of men. The poorest, the least regarded, the most disposable members of any society are always women. So I do think the term still applies to the UK and practically every country on earth, sadly.