This will not cause data integrity issues, but a server may confuse requests coming from different sessions in the same browser. This is not the Aware IM problem - it s a general problem with all web applications that use cookies to distinguish between sessions.
I believe this issue has already been discussed on this forum several times.
You do need to manage this within one application or ask users to logout and login.
You may choose to ignore the issue and if you are lucky you may get away with it.