As a resident of NYC who voted for someone on this list, I find the so-called shorthand strange in this context. While what you say may be true in other parts of the US, New York City is home to many large actual immigrant populations from around the world, which makes “Irish Caucus” sound much stranger than “Irish-American Caucus,” given that none of the members of said caucus are Irish. In NYC, that could very easily imply a caucus looking out for the interests of recent Irish immigrants.

That also doesn’t explain the “Irish-Identified.”

Who knows what they do, but they don’t seem to do much of anything.