In my opinion, the term "institution" simply implies that marriage is and has been recognized by society as valuable to society as as such institutionalized. Society did not create marriage it simply incorporated it as something recognized successful. In the moral free market, the marketplace of ideas, some ideas are rejected and some are accepted, some are but novelty or fad while some become institutions.
Institutions come not from think tanks, elite academics, or government mandates.
Government did not create marriage. Government simply carries out the wishes of society by recognizing and accomodating on society's behalf what society deemed valuable long before government was created by society.
As to religion, from my perspective, marriage entails a commitment before and between God, husband, and wife with religion simply God's representative much like government is simply society's representative.
With homosexual marriage we see government place itself over both God and society. Not good.
That is what I thought... but you put it to words. So thank you for that and have a great Christmas.