In the UK, JK Rowling, even though she is richer than the Queen, is not an aristocrat. She is, in fact, upper middle class. Someone who's born into the aristocracy, even if they're not especially wealthy, is an aristocrat.
Aristocracy always begins with a way to define the elite. Over time social mobility will decline and people will start to notice that it's mainly the children of elite who become elite. Eventually this will lead to the belief that only the children of elite can be elite. Once this belief is entrenched, it's natural for the new reality to be codified in law.
Do remember that even in England there was social mobility. If you performed a great service for King and Country you would be granted land and title.
Social mobility exists in all societies. Just some more than others. In America we are observing the emergence of a natural aristocracy - a story as old as history itself.
The historical exception is the U.S. from 1945 to 1973, as well as perhaps a few other examples (maybe Scandinavia today, or the U.S. in the early/mid 1800s?). In that time period - really, no more than a generation - we saw a big leveling of income distribution, where people born to modest families suddenly had the creature comforts that a generation before had been reserved for the aristocracy. I think most people would agree that this is a generally desirable state for a society, so it might be worth asking what was different about those time periods that made them the historical exception?
Entrenched aristocracies happen when you have old money that looks down on the "nouveau riche". Every country has an old money class that considers itself more elite than the nouveau riche, of course, but they aren't really an entrenched aristocracy unless the rest of society agrees with them.
That's not the cause of aristocracy, that's the effect.