In a web-application implemented in java using JSP and Servlets; if I store information in the user session, this information is shared from all the tabs from the same brows
I think what you probably want is to maintain navigation state across tabs and not specifically creating a single session per tab. This is exactly what the Seam framework achieves with their Conversation scope/context. Their implementation relies on the fact that a conversation id is propagated with each request and creates the notion of a conversation on the server side, which is something that lies between a session and a request. It allows for navigation flow control and state management.
Although that's mainly aimed at JSF, have a look and check if that's something where you can take some ideas from: http://docs.jboss.org/seam/latest/reference/en-US/html_single/#d0e3620