It's a reference to social structure. One imagines a family that is part of the aristocracy, the elite, part of "society."
I think it might refer, not merely to the aristocracy, but specifically to an "old money" family, one that can trace itself back to the early Dutch settlers in "New Amsterdam" (an earlier name for New York). There was a group of families who were scornfully called the "Knickerbockers" that were among the power elite during the Gilded Age of the late 1800s.
Examples might include families with names like Beekman, de Peyster, Van Rensselaer, Rhinelander, Stuyvesant, Schuyler and Schermerhorn. The name "Vanderbilt" is obviously Dutch, too, but I have an idea that they were interlopers and "new money."