He was a wrestler of some renown in his youth and his coach nicknamed him 'Platon,' after his size or big shoulders (depending on who's telling it). Winning arguments by flexing is apocryphal, as far as I know.
It also doesn't really sound out of character, in a general sense. It's not like being an ancient philosopher also meant you didn't have a sense of humour (assuming the flexing was humour, if true).