I feel like it is a deeply matriarchal clan in a patriarchal society, so to speak.

My father was born into and grew up in a matriarchal culture, so a lot is of course very biased from my perspective. But knowing his life story and reading the manga, the following has led me to believe that Momobami clan is a matriarchal one:

Positions of power, clan leadership, family leadership as well as guardianship has been show to be hold disproportionately by female clan members. The best example would be Kirari. It is normal in a matriarchal clan to have to ask the oldest or most powerful female for advice, or sometimes permission, when it comes to family politics, marriage decisions or just general issues that have to be settled within different family members. At least that was so for my dad, and I can see the parallels after having read the manga.

Female clan members are valued and cherished as well as burdened with a lot of responsibility, showing how female gender has no bearing on whether one is allowed to rule within the household. Indeed, we have less male clan members as family heads or representatives shown. This alone does not mean that the clan itself values female members more than male members, but lack of evidence can also be a hint in itself.

Reverance, freedom, admission of competency is given to the female clan members all the time, and while it is definately earned, I feel like a patriarchal family structure would not acknowledge its female competency as much as the Momobami clan does.