When the average guy says he’s a “made man” what does that mean?

is an expression meaning that said man has made his own way in life and has become successful as a result.

Of course. We don't all measure 'success' the same way so be wary of any man claiming to be so.

Generally means he has made his own way, was not born into a rich family or given a position in his family's business

Usually it means you are a "made man" in the mafia and have committed a murder for the cause, those guys don't normally go around telling ordinary people that.