The Irish culture and the American culture are very similar cultures. The Mexican/Hispanic culture and the American culture are very dissimilar cultures.
That's not what a lot of early white Americans thought. After all, the Irish, like the Italians and Poles, are Catholic and religion used to matter a lot more then than it does now, too. You should look at political cartoons from the 1800s and how Catholics were depicted. My father and mother almost didn't get married because my father refused to promise to raise me Catholic and my great grandparents on my father's side are burried in seperate cemetaries despite having 12 children together because one was Catholic and the other Protestant. If you look at the draft riots in New York City during the Civil War (the racial aspects glossed over in Gangs of New York), the reason why the Irish immigrants targeted blacks (including setting fire to a black orphanage) is that they saw the blacks as competition because the Irish were widely considered lower on the social scale than the blacks were.
I do understand the point you are trying to make but you should take a serious look look at how the Irish were perceived at the time and why, nor should you ignore the Anglicization of Ireland and how that happened.