In Mario Puzo's The Godfather, Vito Corleone does not have multiple personalities so much as a hybrid identity: a fusion of two contrasting cultures, languages, and roles within his family and the crime family.
As you know, a hybrid is a sythesis or complement of at least two distinct things fused into one. A hybrid car can run on gas and electricity. A hybrid bicycle can travel on the road or the trail. So it is with identity: since we all are born of two different people from two different families, we are a hybrid of our parents, certainly genetically but also in terms of personality.
Immigrants especially develop hybrid identities. Vito Andolini is the son of a father who refuses to pay tribute to the local crime boss in Sicily. After his father and older brother are murdered, he too is targeted. So, he is smuggled to America for protection. Renamed Vito Corleone at Ellis Island, he is essentially raised by America (he sees the Statue of Liberty from his window). So, he becomes a hybrid of the old Sicilian values and the new American culture. He had spoken only Italian as a child, but when he is quarantined in America he refuses to speak it. Soon, he adopts English as his tongue.
Vito's hybrid identity is a by-product of the old Sicilian values in Hell's Kitchen and the new opportunities that the American dream offers. Like his father, Vito is a victim of the local crime boss (Don Fanucci). Instead of running away from the problem, Vito murders him, thus destroying the old power structure and enabling him to fill the void.
So it is with his olive oil business: it is both a front for his other illegal ventures and a symbol of his family pride. Later, Vito will return to the old country to kill the old don who killed his father, using his new world business status to settle and old world vendetta. All in all, Vito is a hybrid of two things: his father's death and his mother's hope, his Sicilian roots and the American dream, a victim of the old dons and an opportunist to become a new don himself, an honorable man and a criminal, a loving father and a destroyer of families, a Godfather protector and a murderous vigilante.