not just anarchists, feminists also should balk at sincere efforts at matriarchy, since gender equality is the goal of feminism, not gender supremacy

though I do think sometimes what is meant by matriarchy and what women dream of as matriarchy is more of a balancing away from patriarchy rather than recreating it with women at the helm (or even re-envisioning it if not recreating), that's just not my read of these kinds of memes

it's not really surprising lay people would have less literate and sophisticated articulations of their gender struggle - it's important to read the context carefully, women and men struggling under the conditions of patriarchy often look quite different than ideal and academic articulations of liberatory politics. The default seems to be reaction - e.g. some men unhappy with patriarchy blame women for the problems and fester into cultures like incels, pick up artists, and the rest of the diverse mannosphere. Are we really that surprised women struggling with patriarchy have imperfect articulations of that anger and victimization?

It's all the systemic and cultural things that propagate gender-based hierarchies, usually with white men at the top. It's a concept used frequently in feminism. https://gender.study/gender-based-violence/understanding-patriarchy-feminist-perspectives/ has a short intro (you don't have to read the critiques part if it's not interesting) that covers some different aspects. There's a lot that can be placed under the umbrella like reproductive rights, gender norms, domestic violence, and toxic masculinity. Even though the patriarchy is theoretically advantageous to those placed at the top, it harms everybody.