I see multiple alerts for the same aircraft if I am at the computer, but they only log once in the alerts log.
I have not had good luck with the JSTARS double filter, so I leave it off. It is supposed to keep the same aircraft from being listed twice within a short time period. In my experience, sometimes it prevents the aircraft from being logged at all.
Also, the second hit on a military aircraft is usually the one that carries the callsign.