No, Brainshark does not support different login sessions in multiple tabs in the same browser. This is caused by the mixed protocol that occurs when multiple sites are open in the same browser.
As a workaround to this you can use either an Incognito Window in your browser or open another browser to have multiple instances running simultaneously on one machine.
To request this functionality, you can visit our Product Suggestion forum. To access the Product Suggestion forum, please sign into your Brainshark account and select "Product Suggestions" from the drop-down menu in the top right corner. This will automatically create an account for you if you don't already have one (and if you do, you will be signed in automatically).
In our Product Suggestion forum, you can submit requests and vote on suggestions other Brainshark users have made, as well as subscribe to them for updates. This lets our product team know what features users would like to see in the future.
If you need additional assistance, please contact Support.