It's more of a case of who was defined as "white". Benjamin Franklin once warned Americans to be vary of "swarthy" people, like the French and Swedes. Irish people weren't considered white initially and it was common for places looking for workers to display the sign "Irish need not apply". In turn, later Italian immigrants had to build their own churches because they, albeit Catholic, weren't allowed to use Irish churches.

Jews were also not considered white, and establishments that didn't allow Jews were labeled "restricted". More recently, Hispanics are more and more considered to be white in America, which they weren't previously.

Historian here, I would agree that it’s not going to be as simple as looking at ethnicity. You also need to look at region and particularly at economic competition. For example, there were Irish immigrants and free Blacks in antebellum New York City who shared neighborhoods, saloons, and intermixed culture (where tap dancing comes from), but you can also find wealthy Blacks hiring the Irish as servants in Boston and looking down on them, and you can also find virulent, ugly racism in Irish-American communities made visible in the Boston Busing Crisis, for instance.

I like your question and bangdazap's reply, but I find that there is some confusion about the use of the term "racism". As a child in the 70s, I remember the famous PSA of a boy asking his grandpa what prejudice is. It seemed to me that racism was just one form of prejudice, and did not include prejudice based on religious beliefs or ethnic grudges among different people of the same race.

As far as prejudice among different white ethnic and religious groups, the answer is that it was very common. However, it is difficult to say precisely when prejudice based on religion and ethnicity becomes actual racism, based on biological/genetic differences.