I heard somewhere that it was possible to use PHP via the Ajax Object.
So you could send an request through the Ajax Object to your Webspace and it will check who you are through a PHP Script. The Script can then return a Session ID or other Data(PlayerNames, etc.) which you could try to associate to the current Browser the Player is using. You would have to set a cookie, maybe through Local Storage? That might be the Thing what you are looking for.
But I don't know anything about PHP at the moment so that is all I can do for help. ^^'