The question is rather unclear. How exactly the client should manage info about who he is, if that is just to display on players screen "I'm Mike" or to send out commands "I'm Mike, moving knight from c5 to e6" ?
When client connects to a server, it says who he is during login. Then both client and server have that info. Since servers can't trust clients, the server must manage list of clients and know which one is which when sending out packets/messages.
Servers can't trust clients, to disallow kind of cheats, when client named Jack sends a message "I'm Mike, moving knight from c5 to e6", hijacking the game.
Some game messages are addressed and others are broadcast to all. Being able to send messages via server to a single client, to a set of clients or to all at once is a right kind of design.