Patriarchal societies are ones where elder males are the heads of households (generally true) and descent (such as surnames) are passed through the males (generally true).
There are matriarchal societies – including many Native American ones. Those are called matriarchal societies. They're notably different from what we have. I think they're less common today than they ever have been, but patriarchal societies are also "less patriarchal," i.e. males don't own females, as you stated.
Feminists, and everyone with an education, use that term because we do live in a patriarchal society.
Edit: In response to your comment below, you're literally in the comment section of an article discussing the repercussions of men "failing" in their expected social role of head of household.
Women are surnamed Kaur (princess) and the men are surnamed Singh (lion).
Patriarchal and matriarchal?
Today's society is one where men make most of the laws, own most of the property, earn most of the money, commit the most violence, set the behavioral norms and so on. Everything of importance is controlled by men, white men in particular. Using the word patriarchy to refer a society like that makes perfect sense to me.
The only valid point is ownership of property, and I agree with it - although I don't want it to be more fairly distributed across sex, but rather across the whole population (the fact that owners are the 1% gives them much more power than the fact that they're white).
Except it's not true: http://www.businessinsider.com/women-now-control-more-than-h...
Women do not accumulate wealth at an equal rate to men, one reason is that their salary is much lower (wage gap). A second reason is that most single parents are women and that pushes the wealth distribution for women further down.
In terms of ownership it's closer to 36%.
http://gender.stanford.edu/news/2011/shortchanged-women-and-...
It's especially bad for unmarried women who have about half the wealth of unmarried men. When you take race/ethnicity into account the gap is even wider.
http://globalpolicysolutions.org/wp-content/uploads/2014/10/...
In other word, she already owned half his property, he just didn't realized it until she divorced him. Hahaha.