There isn't a formal aristocracy and there never has been one. There certainly is not a clear ranking of aristocracy with royalty and nobility and such. Money is more important relative to background as compared to Europe, certainly to Britain. There isn't class privilege like it Britain. You don't need to be from a certain background to be a military officer or work on Wall Street. There are plenty of wealthy people not from fancy backgrounds. There are also certain fancy neighborhoods where you sort of have to be from a certain background, and others where no one cares.
There was a pretty strong class system in the old south, but land and slaves were property that could be bought and sold. In colonial times there was an oligarchy that were members of the governor's council. There is the social directory, and people in it are supposed to marry others in it. Presidents Roosevelt and Bush were in it. They may consider themselves aristocrats, but it isn't respected as in Europe. Ordinary people don't care about that sort of thing, and aren't impressed by it. There are elite clubs and elite private schools and so on. There is the Society of Cinnicinnatus, with one member for each Revolutionary War officer, along the oldest male line if possible. Senator McCain's father and brother were members, as was Winston Churchill through his American mother. Americans who made huge amounts in business would marry someone classy and then try to marry their kids to British aristocracy.