What you're asking about Europeans is like asking an American who lives in southern California, if she feels a sense of belonging when she travels to southern Georgia?! She may never have left her own country but she would hardly feel a sense of belonging in the American south! Even a guy from Ohio would get the same reception from his so-called 'fellow Americans' if he wandered into the American south.
Europe is no different, though Europe is a continent and the USA is a country.
Each country in Europe is as far apart from each other as are the American States from each other and many European countries are no larger in size than the average American State.
The difference is, in Europe, every country has their own distinct language whereas in the country of the USA, everyone speaks one language which is American English, which is a derivative of British English and British English is a derivative of European English from historical times.