What shared heritage do the Irish and Italians have that the Irish don't also share with the (overwhelmingly Catholic) Mexicans? They don't eat the same food. They don't speak the same language (in a linguistic sense, Mexicans have more kinship with most of Europe than the Irish, who preserve a Gaelic tradition as well). They don't listen to the same music. They don't live in the same places. They don't grow up hearing the same stories --- outside of the biblical stories that they share with the Mexicans. They don't have the same kinds of governments.

Apart from a shared complexion --- a complexion, by the way, that we used to dispute was "white" in the case of Italians --- what ties a Irishman to an Italian?