Shortly after she left the show CK made a bunch of seethe posts on Twitter and one called him out for having his parents pay for his entire schooling, so he came from money but I think they basically cut him off after his education to figure it out on his own. That degree combo also explains why he has to work at restaurants, I doubt a bachelor's in biology is very in demand, let alone broadcasting. He's pretty much just qualified to be a teacher, and I struggle to imagine Brian teaching.
He probably cries about being poor so much because he never experienced it until adulthood and he's not used to living that way, even though he's still better off than plenty of people thanks to his background.