The rich and entitled (literally) make odd choices all the time in the real world, so why wouldn't they in a fantasy setting? Seems quite plausible to me that the scion of some family might go off on some adventures before settling down. The family might give him their blessing or it could be a source of friction.
As for the bodyguards, in a game like D&D without mechanics for handling allies in chargen, that's a setting question for the player and DM to sort out (ideally in consultation with the full group). You can easily justify it either way. Plenty of people who stand to inherit don't travel with bodyguards. We're talking a minor duke after all. Moreover, it's perfectly reasonable to imagine that he escaped from his bodyguard because he didn't like them or found them stifling or whatever. That adds a fun plot thread that the DM could introduce at some point... running into one of the bodyguards or another agent of his parents who want him to come home.