Marriage is viewed as a social institution because that is precisely what it was in the 19th century. In those days women would enter marriage by offering a dowry in the form of money or properties to a potential husband. Often families did this with other families who had lots of money. If you didn't have it, your chances of obtaining any benefits from marrying money and property became lower and lower. This also meant that your social standing will come tumbling down: You would not be admitted in other homes as a visitor, people will talk about you, and you may end up joining a nunnery or as a spinster, which was worst than being a prostitute even.
In the case of Ernest when Lady Bracknell sat down to ask him all sorts of questions about his finances, upbringing, etc, he mentioned the fact that, although he had lots of money, he was orphaned and abandoned in a handbag. This was problematic because, like Lady Bracknell said, her daughter cannot be married off to a "parcel" and that Ernest needed to produce a father or a mother quickly to be able to consider his engagement. This was the way marriage was done: With names, last names, amounts, and business transactions that will ensure that the family name and money will continue to be long lasting.