Here's a link to a thread from a few years ago, with a good back and forth between /u/nhnhnh and /u/Ruire (amongst others) about this subject. You are right to observe that racism/discrimination was not always linked to skin color. As you will see in this thread, anti-Irish sentiment was rooted in the religious (anti-Catholic) and other cultural differences (language, lifestyle, etc.) between Britain and Ireland, as well as the frequently rocky relationship between the Irish and the British empire.

So was the American bias against the Irish just a continuation of British religious bias? Or were there uniquely American issues? I am mostly wondering about the period from around 1840 through World War 1.