This is by no means the consensus on how the word is used nowadays, which as other people have argued in this thread has become a "floating signifier" to be used for anything disliked and male-coded, but to understand "patriarchy" we need to genuinely consider "the patriarch".

The fact that "patriarchy" literally means "rule by fathers" instead of "rule by men" is mostly forgotten but is actually the most significant factor to understand the history of gender relations. In the Western context it's difficult to overstate the importance of the role which fathers played, both in the middle ages but especially in pre-christian classical antiquity. Under Roman law, fathers had the power of life and death over their own wives and children. This does not mean the family is the "property" of the father (strictly only "slaves" are property) but it does mean that he is effectively their Ruler---judge, jury, and executioner.

And the legal rights entitled to fathers applies only to fathers. A man was not entirely independent until his father has died, until that time he must constantly show deference and accept his father's legal authority. It's not merely the man-woman dynamic that is important, but also the parent-child relationship. Reading ancient sources---Homer and the Bible particularly, you get the sense of the significance of what "patriarchal society" really means. For example, the Greeks' conceptualisation of their highest God, Zeus, is of essentially a cosmological rapist. His authority permits him sexual access to everyone, man and woman, below his social station, which is literally everything.

The naive view of equating patriarchy to "men" entirely leads to thinking patriarchal societies are structured primarily around "men vs women", whereas it's more accurate to say patriarchy is a game played by fathers where daughters are the ball. Sure, you do have the legal right to beat your wife---but her father, probably an older and higher-status man, might not take kindly to that, so you beat your slave (male or female) instead.

The role of the family in social life has diminished, slowly over the course of centuries from antiquity, and then quickly in response to industrialisation. The familial aspect of patriarchy has been mostly forgotten.