For two summers during high school, instead of joining her classmates at the beach, Noura Ghazoui had an internship at the town hall of her hometown, Borghetto Santo Spirito, on the Ligurian coast.

But when she tried to apply for a job there at age 19, she found herself ineligible because, like hundreds of thousands of children born to immigrants in Italy, she could not get Italian citizenship.

“I feel Italian, I think in Italian, I dream in Italian,” Ms. Ghazoui said in Ligurian-accented Italian. “But I am not recognized in my country.”

For generations, European countries have used mostly bloodlines to determine citizenship. The United States was an exception in the West as one of the last countries to grant citizenship unconditionally to virtually anyone born there.

President Trump’s order seeking to end birthright citizenship for the American-born children of undocumented immigrants, which a judge temporarily blocked last week, would bring the United States one step closer to Italy and other European countries.

But rising numbers of migrants in the United States and Europe have set off debates on both sides of the Atlantic over whether the systems for bestowing citizenship need to be updated in some way, either moderated or stiffened.