The Greek thing is ridiculous. My parents come from two different continents, and I was born and raised on a third. My dad was the son of immigrants himself, which throws another country into the mix.
I lived in a few places growing up, and now I am settled somewhere completely different. It means that my background is quite complex to explain (and actually extremely triangulatable, because I have a lot of rather strange countries in there!). I don't have much of a concept of "home"; anywhere I go, I'll be foreign - although I feel very welcome and happy where I live now.
Ultimately, though, I'm the product of my own experiences. My granddad left his home country to work on the railroad - I don't feel a rush of genetic nostalgia when I get on a train.
Jack has lived all her life in the same town, apart from when she was in London with Allegra. She speaks with an accent that is undeniably English. Even her wistful talk about her Greek name...Melissa is a lot less obvious than, say, Ianthe. Oh well. I suppose her Greek phase is less offensive than when she was torturing cats.