European ID cards
If the Swedish or German government inroduced an ID card, would I trust them to make individual privacy a top priority, and to not subvert the scheme into a means of tracking every detail of their subjects' activities? Possibly. Would I trust Amber Rudd with the same question? Absolutely not.
Germany does have ID cards - they're administrated by the Länder, the 16 federal states, but are now printed centrally. The Länder are on strict instructions, that should the Federal Government become authoritarian, they are to destroy their records. The reasons for this should be obvious to all.
Sweden also has ID cards.
The EU has a list of all identifying documents issued by each member. However, it doesn't indicate what can be used if the police ask you for ID. Britain is one of the few countries that accept your driver's license, which doesn't indicate your citizenship, only which country it was issued by. However, the police will check with the Home Office if they have doubts over a persons citizenship.