In this case, aristocrat was used sort of a way of saying 'snobby'. Basically he was suggesting she felt her Betazoid half was superior to her human half (in the same way someone may, if one parent were from money and the other weren't, only concentrate on the former heritage and look down on people who were poor). However in Troi's case she is actually aristocratic to an extent. Lwaxana holds an official title which means her daughter is, even if not personally titled, exposed to that way of living.
Aristocratic titles may or may not be hereditary. For example, Britain overhauled its peerage with the Life Peerages Act 2005, so that it now allows a recepient to hold the title and sit in the House of Lords; however, they do not pass this honor on to their children. So under a system like this, Deanna may not inherit any title. Or perhaps Starfleet requires anyone entering the service to swear away any titles of nobility or peerage. And Betazed may have its own rules regarding the passing of titles from one generation to the next.
Marina Sirtis said once in an interview that her husband thought that Troi's spiteful and overbearing attitude in this episode was the first time the audience had seen the true attitude of Marina Sirtis. Personally, I prefered her "attitude" in "Man of the People". Admiral William Thomas Riker 17:37, August 3, 2011 (UTC)