I would assume anything that actually did this was 1. having a serious bug. 2. Had somehow gained improper security access to the browser and should be investigated as to what is going on because it sounds like, super-insecure!
however I would also assume that I was wrong somehow and what I thought was going on couldn't be it. What have you done to ascertain that the browser really needs to be closed in order for Office 365 to allow you to re-login as another account?
