I think they should be able to claim that ancestry, because I promise- children born in the U.S. to immigrant parents do have a "different" upbringing.
One of my grandmothers (mom's mom) was from the Czech Republic, the other (dad's mom) was from Australia. They raised their kids a bit differently than the typical US family. My parents don't have southern accents although they were born in the south. They cook differently. They wore different clothing than most people around them who'd been born to native-born US citizen parents. They had different mannerisms and beliefs. They really never fit in with their peers, and some of that spilled down to their kids (me and my brother) as well. DNA memory is a real thing, as is cultural upbringing, and I think my family were a little bit more "European" than the families around them as a result of those.
*Note: although this is all true, I would not go so far as to say "I'm European" if I introduced myself to someone, nor would I pretend that I know all about the cultures my grandmothers were born into. I'm only saying, they were not raising their kids quite the same as a native-born US citizen and therefore their kids didn't really fit in seamlessly with US culture.
