Use persistable sessions instead of context

[dokterbob]

Is your feature request related to a problem? Please describe.
A lot of information regarding the user's session is currently stored in context variables. This means that all of this information is currently stored on a smaller-than thread level. That means that our chainlit daemon is stateful -- it's basically stored 'on the websocket'.

This effectively prohibits running multiple instances of chainlit behind a load balancer, as any reconnect will cause the session to be lost.

Describe the solution you'd like
Maintaining state explicitly and reduce state wherever possible.

1. Reduce state in API: Some parts of our API are stateful (actually, it seems, quite a few). Perhaps some of them don't need to be.
2. Switch chainlit context to use 'proper' sessions, e.g. as they're normally used in a web context. Perhaps something like fastapi-sessions.
3. Bonus; strict typing for context's properties.

Describe alternatives you've considered
The alternative is to maintain state fully on the client side and remove all state server-side -- so each request sends the full state to the server. This is more REST-ful but it's a huge leap and requires careful assessment of security implications.

Something like a cookie signed by the JWT.

Additional context

Related issues:

• Screen Output Stopping Issue Due to Lack of Load-Balancing in socket.io Sessions #1492
• support socketio auth option in connect handler #1497

Related resources

• https://medium.com/@pradoshkumar.jena/session-based-authentication-with-fastapi-a-step-by-step-guide-ca19e98ce0f9