I think it is entirely possible, given the nature of their business, that "The Five Families" is their way of saying "the New York mafia", but without direct reference. I believe that may be the case, in the same way that they never use the word "Mafia" or even directly refer to themselves as an organisation at all. Historically it has helped them to pretend that they don't exist, and the tradition still continues to a much lesser degree today.