This is the use-case:
My Question:
How should server B authenticate the user?
Note: server A is a node.js, express server B is socket.io dedicated server.
Just use one of the many non-in-memory session stores. Some example session store Express/Connect middleware include:
If you are creating a SPA (e.g. with angular.js), you could also use tokens (e.g. JWT) instead of using cookies to track sessions.