Thin privilege is being able to play a video game and have your character look like you. Either the default characters have thin bodies, or the RPG you’re playing — despite having hundreds of character customisation options (height, musculature, bust size, even eyebrows) — still only allows you to play a thin person.
Apparently only thin people can be heroes.
I’ve seen a few online flash games with fat people as the default character, but when that is the case, the title is always something like “Fat Guy Adventure” and “fat” is the character’s only defining characteristic. The goal of the game is usually something offensive like trying to collect donuts.