This is definitely super annoying. I think you may need to try to just avoid the topic if you want to save the friendship - you'll get too irritated otherwise. Perhaps it's just a phase she'll grow out of. I don't really think confronting her about it will go well.

Incidentally, I'm from New York too and grew up with a bunch of Irish-Italians and Italians. I'm a little surprised at her, because one very common quality these families have is a deep pride for their heritage, particularly the Italian side. I can understand someone from maybe a more bland Caucasian background envying your strong ties to your culture and community - for example I'm from vaguely British and Western European extraction, and my family's been in the States long enough that we don't really have a community or any traditions or culture behind the same boring old American that everyone shares in in this country. But I wouldn't immediately think an Irish-Italian would feel this way. Maybe you could occasionally ask her about her background and what she thinks of fondly from her childhood and her family's traditions - maybe that would remind her she had stuff to be proud of that are special to her heritage, too.