I heard on a TV show yesterday, and a number of websites confirm, that the idea of corned beef and cabbage as an Irish food is a purely American invention, unknown in Ireland.
Apparently, the Irish immigrants who settled in the Lower East Side wanted to make boiled bacon and cabbage, which is the traditional Irish dish. But they had no real choice but to frequent the many kosher butchers and delis that were owned by Jewish immigrants.
Obviously there was no bacon available at these shops, so they substituted corned beef from the kosher delis, and the rest is history.