I notice it more in families that are relatively recent immigrants, or those that live in a community where everyone is originally from a specific country, and they marry from within that community. So for example, if someone's parents were from Italy, or maybe even their grandparents, they might refer to themselves as Italian or Italian American. Alternatively, their family could have been here for several generations, but they live in a community that is heavily Italian or Italian-descended immigrants, and therefore identify themselves as Italian or Italian American.
Personally, I'm an American. No prefixes or modifiers, partly because the paternal line on both sides has been here since before the revolution, and also because I'm a mutt and it would take too long to list them all in normal conversation. Unless it's normal to introduce yourself as a mostly Welsh, German, Mexican, French American, with a dash of Irish and Spanish, and I somehow just don't know it.