1. (in British usage) illogical, back-to-front. (This term is used by prejudiced people.)
2. (in American usage) not really Irish at all, lacking Irish citizenship, having little or no knowledge of Ireland, and knowing nothing of the Irish language or Irish culture, but possessing a great-great-grandparent (or the like) who was Irish. Not regarded as Irish by actual genuinely Irish people (those with Irish nationality).
1. "Why are you doing it that way? That's so Irish!"
2. American One: "I'm Irish."
American Two: "I'm English, Irish, Scottish, French, and German!"