Session scope between browsers is not possible. Sessions are identified by a token, which must first be given to the user, and then passed back later by the browser in some form. Generally this is done with cookies, although it can also be done by appending the token to URLs as the visitor browses around the site.
Since web browsers are separate pieces of software with their own methods of handling cookies, you cannot share cookies between browsers, and therefore you cannot share cookie-based sessions. It is possible to copy-and-paste a URL from a web site that contains a session token into another browser and continue the session there, but most sites use cookies, so this is not often possible, and it certainly doesn't accomplish what you would like to do.
What you ask is generally considered impossible, but also usually not an issue. On the plus side, it is also a process generally understood by most users. Users do not expect to log in to a site with one browser, and then boot up another and still be logged in.