Forms Authentication with cookies (regardless of whether they are persistent or not) is browser session based (persistent cookie would of course work across multiple sessions of same browser (on same user account on same machine). So two browser sessions (or two different browsers or browser on two machines etc) would be treated as different scope as far forms authentication is concerned.
So user can make multiple login from different browser sessions and logout in one will not affect other. Its is up to web application whether to allow multiple concurrent logins for same user or not. For example, online banking sites would restrict to only one user session - so if user logs in from different session then earlier session is invalidated (i.e. user is logged out). One has to write custom implementation in ASP.NET to do so - typical implementation would make every user session entry into database (typically required for audit purposes anyway) - so whenever new entry is added, a check is made to see if there is any active session for same user and if yes then that session is marked inactive. Every request would check if current user session is active or not, if not then it would flag such message to user.