After testing, I've seen this doesn't work on every server.

When connecting to a server you're already logged in to with another client, the first client says "You are connected from another location!" while the second client connects as normal, but you'll see "<your username> left the game".
Other players will see "<your username> joined the game" and "<your username> left the game" is immediately followed.
When opening the player list (default tab), you won't see your username an other players won't either.
Also, player selectors won't work on you any more, until you die. Commands with actual names in them do work.
When you die your skin also gets reset to Steve? or Alex.
The glitched player is still able to do everything a normal player can.

You may see with the player heads in the tab list included as attachment I'm not there.

I will do more research in the future.