...click remember me, open up a new tab then close the original one, load the web site again and you will be logged in automatically.
What you describe above is very different from the standard "remember me" functionality that I am used to with most websites. The standard functionality that I'm used to is, after you click "remember me", the browser will remember things like your user ID, or remember that you are logged in. You can then completely close the browser, go away and come back the next day, relaunch the browser, navigate to that website and it will remember your user ID but not your password or remember that you are logged in and you will remain logged in. This is done via stored cookies that are website specific, but these cookies are not session type cookies. Stored cookies are persisted to disk and reloaded the next time you load the browser and navigate to a specific website that stored that cookie in the first place. It has nothing to do with remaining in the current session.
In contrast session cookies exist only for the life of the current session and are discarded when you close the current session. Unfortunately Test Studio is not capable of launching a new browser window and sharing existing session cookies. The launched browser window will load the stored cookies after you navigate to that specific website. Thus Test Studio cannot handle the test case you describe above, opening a new tab and sharing the session cookies with the previous tab. But I honestly question whether or not that's really necessary.
So after that rather long-winded explanation, does your website really behave differently from the standard "remember me" functionality that I described above?