A worthy topic for my first ever post !
The one obvious factor is that I & II are prime examples, not only of great films, but all-time classics. However, in order to get so deeply into the public's psyche, the film had to have themes were at the outset both appealing and empathetic.
The appeal comes from the rather glamorous interpretation of the Mafia. Perhaps this wasn't realistic, but that actually helped as what we saw was an almost Medieval-like tale of power, intrigue and betrayal in a modern, but deadly setting. The films, despite drawing on disturbingly familiar and actual events, were in reality set in a fantasy realm.
The empathy comes from the fundamentally strong theme of family that pervades the whole span of the 2 movies. Often such films struggle to appeal because of their tendency to dwell on the mundane. But here we see the familial displays of love, loyalty and betrayal that we all know and have often experienced, but within a glamorous and dangerous setting. And because the viewer is taken so deeply into this realm, it flatters us to see everyday themes that we are so familiar with played out so dramatically.
If you want to go even further, then if you are Catholic yourself, then the empathetic chord strikes even deeper, because of the recurring themes of the Sacraments that are virtual pillars around which the plot is constructed.
Of course, all this can only work if the technical qualities of the movies, i.e. direction, cinematography, plot construction and acting, are themselves excellent, which of course they are. But I think the most enduring quality is that fact that no matter how many times you watch the films (I make a point of doing so at least once a year), you can always discern something new. And that is only possible in a true work of art.