There are plenty of exclaves with country borders out there. For instance, here's a bit of Germany inside Switzerland:
https://goo.gl/maps/bw3PwD19JhR2
The village is called Büsingen.
https://en.wikipedia.org/wiki/B%C3%BCsingen_am_Hochrhein
In United Arab Emirates there is an Omanian exclave called Madha, which in turn contains a UAE exclave called Nahwa:
https://goo.gl/maps/REXtsfb3gHs
https://en.wikipedia.org/wiki/Nahwa
And there's even one third-order enclave.
https://en.wikipedia.org/wiki/Dahala_Khagrabari
Dahala Khagrabari is a piece of India, surrounded by Bangladesh, surreounded by India, surrounded by Bangladesh.
And things stay this way because it is quite awkward to change places where people live to belong to another nation - however impractical these exclaves may be.
It wouldn't be necessary really, as citizenship is more dependent on your parents. Both of them being EU member states also makes residence somewhat irrelevant too, except maybe with taxes.
Are you sure? There are plenty of second-generation immigrants here that have both the Dutch and parental (mostly Turkish/Moroccan) nationality.
https://www.google.fi/maps/place/Switzerland/@50.5972041,6.2...
Scroll a bit further south from the farm, and you'll see a Belgian trainline cutting off (much larger) bits of Germany.
Lots of people would be unhappy to suddenly live in a country where they're not a citizen and subject different laws, taxes, and health care. In fact, it's not hard to see why it would be pretty much a non-starter politically.