They do, all IDs are in ID-1 form factor, stamped by the countries and the EU [1].
It's the same for driver's licenses [2], they are accepted in all EU member states and basically look exactly the same, just in different languages and show a different flag.
[1] https://en.m.wikipedia.org/wiki/National_identity_cards_in_t...
[2] https://en.m.wikipedia.org/wiki/European_driving_licence