How to translate "rapporto bancario"?

In Italian, we use "rapporto bancario" as a generic word for any "banking relationship" you have with your bank, like a banking account, a mortgage, a loan, you name it.

Is "banking relationship" a correct translation?

I would say that, yes, "banking relationship" is the correct term, for financial institutions that accept deposits and make loans -- banks in the traditional sense of the word. During the five years I spent a while back at a large corporate bank on Wall Street, that was the term we used. It is also the term my retail bank uses currently.

Investment banks and wealth management firms tend to use other terms like "client relationship" or "advisory relationship".

In addition to what @IsabelArcher says, there's another level of relationship in which one party has a duty of care, and is required, often legally, to act in the best interest of the other party. This is a fiduciary relationship.