For centuries, the Catholic Irish had marked St Patrick's Day with a special meal at home.
A firm favorite was unsliced back bacon, boiled and served with cabbage and potatoes. If available, other vegetables were chucked into the mix. The whole dish would be presented with a sauce.
Often this would merely be the boiling juices trickled over the top. But if they were feeling particularly flush, the Irish family would pour over some white sauce (a mixture of flour, butter, milk and parsley, though another herb can be substituted).
Bacon and cabbage was thus the closest thing that Ireland got to a traditional special meal for St Patrick's Day. But they weren't adverse to eating it any other time too.
Naturally, given the circumstances from which they'd fled, and now found themselves, this dish was a distant memory. It was certainly not anything that a resident of the Five Points could afford to cook.
So they got creative. There was no bacon in their price range, so they took inspiration from their Jewish neighbors. The Jews had a salt beef, cured or pickled in brine, the preparation for which they'd brought with them from Eastern Europe.
Generally cut from older cattle, hence being quite tough, it sold for a lot less than other kinds of meat. The Irish could afford it! At least once in a while. So that was their bacon substitute. This briny beef brisket is what's today known as corned beef in America.
Guess what was the cheapest vegetable you could buy in the New York slum area?
Voila! Corned beef and cabbages was invented! Not over in the Emerald Isle, but in the middle of lower Manhattan. The people who conceived of it, and popularized the dish, had indeed been born in Ireland. However, they were now Irish-Americans.
Their St Patrick's Day religious observances called for the the best meal they could produce on March 17th. Right then and there, that was corned beef and cabbage. Hence the association stuck and it's still a thing in the USA today.