Assuming the application has multiple sessions turned ON and the same user is able to login via different browser it will also depend on what value is set for EnableKeepAlive. Also, I think that a session is cleared out and not the user when the sessiontimeout is exceeded.
If you need more information regarding this topic, I’d suggest you to check this link.
You can verify the behavior yourself for your application because your settings could be different. But most likely, the user will not be logged out from the second browser as that is a completely different session.