I don't think anything like that goes on in the US. The US has 40 horses and men used for ceremonial purposes, and has used mounted special forces recently. However, traditional cavalry divisions and such are now tanks and helicopters.The Black Watch was fashionable, but it wasn't as prestigious as the cavalry or Household Division. The same still rings true to this day. An old school associate of mine joined the guards (those guys with redcoats and silly hats). I asked him why he didn't join the cavalry and he said he couldn't afford the social life. Cavalry officers are expected to own their own horses and play polo,, as well as paying for their mess bill and whatever other fancy stuff they have to do. Being a general line officer from one of the county regiments still has some swagger, but the closer to the crown you get the posher things generally are. The hussars are fancy too due to their flashy uniform and the charge of the light brigade business.
Re the OP, the Americans may have an oligarchy but I don't think they have an aristocracy, you need a monarchy and titles for that.
There aren't posh military units, and being from some sort of background isn't a requirement of an officer. The second President Bush was in a sort of fancy air national guard regiment with others avoiding Vietnam. Diplomats are mostly from a certain background, as it is sort of needed for the job, which requires knowing how to act and what to say at fancy parties and so on. The US is more practical and less into aristocratic tradition.