Patriarchy is an idea that formed after expansive empires became a thing. Oldest civilisations of world usually were Matriarchal because Mother goddesses were given a huge importance and women enjoyed great privileges. Women in Egypt, Elam(Present day Iran), Minoa (Probably Crete), Akkadian(Present day lower Iraq) civilisations etc all enjoyed equal status with men. But when expansionist ideas grew among early Empires, women were gradually sidelined because men were hailed for their martial prowess. Thus we see the shift in Assyrian times, the first empires, that men shadow women and a truly patriarchal society emerges. While we see Greek and Romans celebrate their women, we don't see it as a matriarchy by any stretch. Even today there exists little tribes those are matriarchal in nature. Patriarchy is something I directly relate to expansion and imperialism.
Coming to animals, Hyenas, Elephants, Whales, etc follow a matriarchal society. The lead female shows the herd it's way. We can see matriarchy in lionesses too in some extent but lions, due to their physical bulk overcome them. Many fish live in matriarchal societies. But it would be wrong to call anything below mammals patriarchal or matriarchal because, these lesser animals follow herd/group mentality (sorry for calling other animal families lesser but it is so in comparison with brain capacity). Reptiles, fish and birds just follow the group instead of following a specific gender. Mammals on other hand, have a clear gender dimorphism so it's logical to assume the crowd recognises a gender over other as superior. I hope my reply gave some sense.
The stronger a light shines the darker are the shadows around it.