CASTEL MAGGIORE, Italy — When 1-year-old Kirsi Bestetti tripped and cut her lip at her grandparents’ house last summer, her mother Elisa Bestetti rushed her to the emergency room, panicky about all the blood.

Once there, she also worried whether the hospital staff would accept her as Kirsi’s mother.

Ms. Bestetti is Italian, but towheaded Kirsi is Finnish like her birth mother, Emmi Pihlajaniemi. The two women have been married in all but name for five years at home in Finland, and each has given birth to a daughter who has been legally adopted in Finland by the other partner.

But Italy does not allow a child to have two mothers. Same-sex couples in Italy are not allowed to marry, to register partnerships, to adopt a child or benefit from assisted reproduction. Within the European Union, such family law issues remain the jealously guarded domain of the 27 individual countries, each with its own history, culture and legal tradition.

On the intertwined Continent, which prides itself on its open borders and a single market — as well as on being a trailblazer in banning discrimination based on sexual orientation, even electing openly gay politicians to high office — the resulting differences are more than symbolic. Increasingly they are leading to practical difficulties in all kinds of areas, like taxes, parental rights and inheritances, as people move around for work, love or just vacation.