There’s not one NY accent. I’m from NY, and that actually sounds more like a New Jersey accent to me, but I grew up around Jews and blacks mostly, until we moved to Queens, where there were more Irish, but mostly at least third generation, and a Queens accent is different from a Manhattan accent. Anyway, you could sound different from the people on the next block in NYC.
But yes, that does sound like an Irish-influenced accent. You have Irish-influenced, Yiddish-influenced, Italian-influenced, British-influenced, even southern-influenced. There’s a whole cohort of people who came to NYC after Katrina who speak a New Orleans-New York hybrid, and then, the Black accent is very southern-influenced (I took a class in college on African American autobiography-- we spent a couple of lectures just on AA dialects), plus, I’ve heard it. I lived in a mixed neighborhood called Morningside Heights until I was 8. That’s just in Manhattan. A Brooklyn (say, Flatbush) Jew sounds different from a Manhattan (say, Lower East Side) Jew.
Where can you still hear that accent? It is fading, I expect, because few people from Ireland move to NYC anymore to freshen it, so it may die out, just like the first generation Yiddish accent is gone. There is no one who solely speaks Yiddish as a first language anymore, except maybe a tiny community in Israel of Lubavitch stuck in the 19th century. But new accents come in, because NYC remains a port city. There are Israelis in NYC now, so you hear that accent, and there’s an NYC Hispanic accent that is distinct, and it has been around, I think, probably less than 100 years.
It’s nice you noticed this. We have a record of the accent. Probably someone, somewhere, could do a master’s thesis or dissertation cataloguing things like this. My mother is a dialectologist, and did her dissertation on Slavic dialects.