This is just me meandering through some 101-type thoughts, organizing how I think about them.
I live in a patriarchal society. Almost certainly you do, too, if you're reading this.
This means that I live in a society where the mental image, the myth, posits patriarchs as the natural leading character of human drama, politics, work, and so on. These patriarchs aren't real and specific persons; it's an archetype, like God-with-a-beard. Abraham and Moses? That area. Patriarchs are the manly men; they have natural authority, they're straight, they're cisgender, they're fully able. Where I live, they're largely imagined as white; in other places, this varies.
The myth affects judgement calls. Men can get listened to more, can get less bonus obstacles thrown in their path when seeking career advancement (their issues are normal for leaders; pregnancy is not normal, because the normal leader is the patriarch, see). Lots of stuff like that. A good part of that stuff is built into institutions, even at the physical level. There have been entire sections of universities that only had bathrooms for men and have needed to be rebuilt or reorganized, because that course of study was for men; it was assumed to be for patriarchs in training.
I say "can get" a couple of places there, because these benefits both apply to and are desired by men that fit or at least appear to fit the model that the myth provides. If you want to be a sensitive stay-at-home dad, you're outside the myth in some respects, which means some benefits don't apply or aren't desired. You've got to fit the myth in all the ways to get all the goodies.
The same myth sets up roles for most others as supporting cast. Women are around to be helpers and wives and sexy rewards for heroism on the part of the patriarchs. The same path-smoothing effect exists - for women who want to fit that model to whatever extent, society is ready to assist and advise them in getting there.
Equivalent processes don't result in equal roles, obviously; they don't even result in equal departures from those roles. When a man steps outside the model of the patriarch, he's typically losing benefits that can largely be stated in terms of social power; he's not taken as seriously, etc. When a woman steps outside the model of the supporting cast, she's typically giving up benefits that can be stated in terms of personal security (and I mean money, health, safety, the whole ball of wax).
Among the strangest effects of this is men who don't fit the model of the partiarch for whatever reason, or even want to deconstruct it - but feel entitled to the benefits of the myth, and especially, to the presence of women as supporting cast for them. At high strength, this the realm of the MRA, the men's rights activist. At low strength, it goes almost entirely unnoticed unless it's pointed out; that's just an argument on "What A Real Man Is".
To end this on a true joke: A lot of the dudes in that last paragraph don't believe in the existence of the partiarchy.