Wrong. I first heard about the Seven Fishes dinner from Italian-American friends in college, and have since seen it described as a universal custom in English-language newspaper articles, some of which go as far as to attach a particular significance to a course or number of courses.
But it's not; in Tuscany, for example, those who observe the Catholic custom of avoiding meat on a Vigil will eat fish, but there's nothing that says one must, and Artusi, who gives all sorts of menus, including one for Christmas Day, doesn't say anything about Fish on Christmas Eve.
In other parts of Italy people do eat fish -- because of the Vigil, and Naples is especially well known for its Christmas Eve fish dinners -- but again, the number of courses and the kinds of fish served depend upon the number of diners and the depth of their pockets. There are no hard-and-fast rules.