Catalysts specify a Servant(and in the case of Assassin for a traditional Holy Grail War can overwrite the Class being a catalyst for Hassans), but if you don't have a Catalyst you'll get a good match for your personality.

For example, if Rin had pulled off her summoning perfectly and there weren't [i]certain other circumstances in play[/i] she likely would have gotten Arturia because they synch up incredibly well.