FFC's fanatical attention to detail is at work here. Latin males traditionally grow moustaches to denote maturity and machismo, and Sicilians and Mexicans still do. Note that Michael never had a moustache--he wanted to be "legitimate" and "American" (n.b., at the end of the novel, Kay takes vows in the Catholic Church. Michael is disappointed because he wanted his kids to be raised as Protestants--'it was more American").
Slicked-back hair was common among gangsters and tough guys--longer hair, curly hair or parted hair was considered not part of the image.