Web Browsers nightmare

There is a problem exist with modern web development. For somebody it may be not a problem at all, but I'm blocked up by ....... and have to face it. The problem is the coexisting of the web server session management and multiply open browser's tabs/windows. Each window should have its own session on the server. When a new window opens, the new, its own session has to be created.
It would be said loud, it's not a problem at all if browser's url contains all necessary information about what actual data is being browsed/edited. But sadly, not in this case.
I wonder, what the W3C thinks about that? They invented the sessionStorage object which is supported poorly (IE, Opera) or is not supported at all (WebKit) by modern browsers.
It seems my vision and needs are in contrary with the browser developers'. For instance, Opera has a feature called "clone tab", which creates an exact copy of an existed tab, of course with the sessionStorage object same values, ruining all my efforts to achieve independency of tabs.
Despite some advantages of the sessionStorage object it does not allow to solve the problem of independent session.
It's pity, to solve the problem it would be enough one simple extension to browsers API - session cookies stored separately in each window or tab.