I can't speak for Aristotle, but there is a strong possibility Plato (or more accurately Plátōn) was a nickname meaning 'Broad' either for his intelligence, his physique, or just for having an unusually large forehead. The sources are not really clear about it. Diogenes Laërtius was a biographer writing many centuries after Plato's death, and he claims Plato was originally named Aristocles after his grandfather. If we don't take that at face value, an argument can be made that 'Plato' was a relatively common given name in Athens at the time. Whether he was born with it or not, he certainly called himself Plato in all his writings, so if it was a nickname he preferred it to his given name. That is entirely possible. How many times do you think Snoop Dogg has ever referred to himself as Cordozar Calvin Broadus, Jr.?