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.