I'm confused. That phrase reads to me as "inherited and thus old money, I was poor or middle class but worked hard and made myself rich the hard way"
In my understanding, you used opposite situations to describe the same people. Can you tell me what I'm missing here?
To get a little more in details wrt to the demographics: I am thinking about lawyers, physicians or researchers which have assets >5 million who had to go through the grind of tough academic programs, highly competitive and skilled professions etc.
I am opposing this to rich heirs, actors, athletes, lottery winners, just-sold-my-pizza-app-startup etc. who like to show-off their recently acquired wealth. This is not a universal example obviously, there are outliers everywhere. Only talking about trends here.
But also, most of the unequal laws and policies in a country are created to prop up the position of such people. E.g. house-price inflation helps only those who already own - or stand to inherit - houses.