Hmmm, well it really depends on how you define it. Regardless of whether there's a formalised aristocracy with noble titles, the existence of political dynasties is nothing new and the US was no exception. Some common terms you might be familiar with which conceptualise these early aristocratic classes would be "Boston Brahmins," or "planter aristocracy," or "first families of VA," "colonial gentry," "american gentry," etc. you can come up with a number of ways to refer to them. You also have to understand that there are many political families that might be well-known locally, but lack the national recognition garnered by families such as the Kennedys, Bushes, Roosevelts, etc. One thing that helps us remember those specific political families are because of their multiple presidencies (there are only 4 families to do so); however, they are only a minority of the numerous political families that have governed America.

In terms of as an independent country, the US has had political dynasties since its founding. The Adams family would be one of the most notable families, producing two presidents famously. 2 of John Quincy Adams' sons would also go on to become state representatives, as would some of his grandchildren. The Adams family also produced many important figures outside of just politics.

But if you look at the history of the US, back to when it was just colonies, there were many prominent families that formed local political dynasties. One name you might have heard of is the Lee family. The Lee family is most famous today for their association with the Confederate General Robert E. Lee, but the family has a very long past (in American terms at least.) Multiple members of the family have been Burgesses and members of other government institutions of Virginia since its founding, they produced two signatories of the US Declaration of Independence (out of the total 56 and 7 from Virginia,) served as president of the Continental Congress, produced multiple congressmen after the foundation of the United States, and held the governorship of Maryland multiple times. If you're talking about an influential political family, the Lee's are one of the most important, even if they did not produce a president.

Basically, since the colonies formed, there have been political dynasties in the US. I just gave you a very brief overview of two famous dynasties, but there are far more. Also, you have stuff like proprietary colonies (i.e. Maryland) where the Calverts appointed governors of Maryland until the American Revolution (outside of a period due to the Glorious Revolution.) I chose the Lees because of their fame in modern America, but you really can't go through Virginian history without studying the Randolph family. The Randolph family also produced multiple burgesses, governors, congressional members, and Thomas Jefferson's mother was a member of the Randolph Family. Essentially, if you read through early American colonial history, the same names are going to pop up again and again.