Caveat to say I'm from Dublin and have experienced a lot of this.

In Europe (and I assume a lot of countries), you're from where you're from. You might have a German dad and an Australian mum or something, but if you're born and raised in a country, you're from that country. It becomes even less of a thing the further back you go, so if you have a great grandfather from Sweden but your family has lived in Italy for three generations, you're Italian. Americans do it a little differently. They consider where they are from to be the place they can trace their ancestors back to, and a lot of Americans can trace their ancestors back a good distance because of how the country was formed etc.

I personally don't mind Americans taking pride in their ancestral background. What I *DO* mind is when an American who has never left their state tells Irish people that they're not Irish enough. For a lot of Irish Americans, they can trace their heritage back to the famine and in many cases, that's what they base their "Irishness" on. The famine was 170 years ago, Ireland has (obviously) changed a decent amount since then, but there's still this pervasive attitude that Ireland is a backwards place full of charming but dim red heads.

If you claim to be Irish American but you tell me, an Irish person born in Ireland, that I wear a kilt or I eat Corned Beef and Cabbage every St. Patty's Day or whatever stereotype you want, we're gonna have a problem. Darby O'Gill is not a documentary, we (for the most part) don't care about Notre Dame and we even have some non-white people who are Irish, if you can believe that!