it would probably be a display name and/or a unique identifier
You do not need access to the profile name for that. I might be wrong about this, but there is a stored cookie and such. Webgames can recognise recurring users. There just is no link to an Itch profile. Also, you do not need to log in to Itch to play web games.
My main point is, you do not interact with other players while playing a game. Not with your account name at least. Itch is a download store, not a social network.
Support for some kind of web games platform login and save games might be a nice idea for the future. But it should be amateur developer proof.