Is "aristocrat" not applied to Americans?

I am not trying to be wholly facetious there is some truth to what you say but such families are often called aristocratic they simply are not part of a formally established aristocratic class. In the sense I think you mean, we generally don't apply the term aristocrat because that is usually reserved for a member of the nobility. The USA has no formal nobility. There is no law in any state granting the Vanderbilt or any other family legal privilege such as can be seen in Europe and elsewhere at the time of the revolution or today.

There were in fact a few aristocrats during the revolution in the United States but as can be seen by the unratified Titles nobility Amendment and Article I Section 9, Clauses 8 of the constitution which prohibits titles of nobility or honor and the next section which applies they same to the states they didn't last long as nobles.

The annotated constitution illustrates the government aversion to aristocracy

The Constitution’s prohibition on titles of nobility reflects both the American aversion to aristocracy1 and the republican character of the government established by the Constitution.2 The Clause thus complements other constitutional provisions—most notably the Thirteenth, Fourteenth, and Fifteenth Amendments—that prohibit invidious governmental distinctions between classes of American citizens.3

The Articles of Confederation4 and many Revolutionary-era state constitutions contained prohibitions of titles of nobility and other systems of hereditary privilege.5 The federal Title of Nobility Clause substantially follows the Articles’ prohibition and was not a subject of significant debate at the Constitutional Convention.6 As James Madison observed in The Federalist No. 44: The prohibition with respect to titles of nobility is copied from the articles of Confederation and needs no comment.7 Alexander Hamilton, in The Federalist No. 84, was only slightly more loquacious:

and again later

Nothing need be said to illustrate the importance of the prohibition of titles of nobility. This may truly be denominated the corner-stone of republican government; for so long as they are excluded, there can never be serious danger that the government will be any other than that of the people.

How broadly to understand the Title of Nobility Clause’s prohibition thus remains an open, if perhaps academic, question. On a narrow reading, the Clause merely prohibits a federal system of hereditary privilege along the lines of the British aristocratic system.11 More broadly understood, the Clause could preclude other governmental grants of enduring favor or disfavor to particular classes based on birth or other non-merit-based criteria.12 Some commentators have suggested, for example, that the Title of Nobility Clause might forbid admission preferences for legacy students at state universities or certain benefits that accompany receipt of the Medal of Honor.13 After the adoption of the Fourteenth Amendment, challenges to governmental favoritism based on class, race, or other bases have usually relied on the Equal Protection Clause.14

https://constitution.congress.gov/browse/essay/artI-S9-C8-4/ALDE_00013204/['first',%20'amendment']

That last part is telling. In American—legally at least—class does not exist so you can't be at formal "aristocrat." In fact for much of American history you could say the term held pejorative meaning (though that may be a sampling bias on my part as I have made no study of it). That being today people often use the term for certain families and the existence of the social register implies that there is an analogous aristocratic class though they rarely go by that name and lack the sovereigns direct backing.