He doesn't. Nobody deserves anything.

Depends on the word games being played. We don't deserve anything, but being born as thinking humans we have certain inalienable rights. They aren't "deserved", they just are.

If the plumber's kid not successful if he goes to school, gets a white collar job and generally lives better than his parents?

I think the point of the question is, that we as a society try to sell the plumber's kid on the idea that everyone gets where they are through hard work and therefore deserves what they have. But the millionaire's kid can coast into a life that the plumber's kid could likely only achieve by incredible luck combined with hard work. In light of that, the idea of the millionaire's kid deserving what they have is called into question.

Is the millionaire's kid successful if he does literally nothing in his whole life, lives off his mommy or daddy's money, and still lives better than the plumber's kid?