Can you clarify a bit more about same browser session ?

A screenshot depicting the issue will be of good help

We have two snap start URL each pointing to a different assignment shape and both the snapstart URL have operators with different access groups for authentication.

Once we open 1st snap start it shows the UI and we perform action and can see different elements based on his accessgroup.

Now open 2nd snapstart in the same browser  which have different UI which is visible based on new accessgroup different from the 1st Snapstart. In this case 2nd snapstart accessgroup is not loaded, it takes the session and accessgroup details from the 1st snapstart which was already opened.

Ideally, both snapstart URL should have opened up with thier own requestor context but its not working that way.

Let me know if that helps or let me know if u need screen sharing session. We can have one so that i can demo you up the scenario. 

Are you closing the browser before opening the second snap start URL ?If not, try that out.