it should not be possible to use a named account more than once at the same time. If you use account A in browserinstance 1, and then log in in browserinstance 2 with this same account A (and actually start working in this instance) then account A should be logged out in browserinstance 1.
So the first case; (the user that was kicked off the server) is how it's supposed to work. I do not know what the desired behaviour in terms of running processes is though.
The latter case should never be possible though, was this in Firefox or IE?
In regard to question whether this behaviour can be changed; not to my knowledge. However, users should have their own named account that no one else should use. If they need to be able to log on more than once they should have two named user accounts