A quick look at LazWebsockets source code I can see that LazWebsockets does not contain any session management features. Well at least I don't recognize any.

This means that you will have to devise your own session management mechanism yourself. You may want to draw your inspiration on how WebServers handle session management.

In short:
When client connects to a web server for the first time it assigns client a new unique Session ID. It then stores such SessionID along with other needed information like client address, client usernam etc.
Then it sends back the client a small package containing the assigned SessionID. The client will then attach this SessionID to each message it sends to server.
This allows server to recognize from whom the message has come from even if client network address might have changed for some reason. At the same time since server stores clients address along with it SessionID it can then send messages to specific clients at any time and not just as response for clients own message.

So yeah I'm afraid you might have quite some work to be done before you will be able to work with multiple clients at the same time. But if I remember correctly you said that this is more of a learning thing to your any way.
So I'm guessing this might be quite interesting learning experience on how session management works and how to implement one yourself.