Yes, this is what happens. National sovereignty still exists. Maybe, since Europeans have a supra-national entity above their countries, they don't quite grok that everywhere else a country is the top giver of the law.
I doubt it’s particularly different to Americans having trouble with the idea that not everyone has the first or second amendments. Or, even, an explicit constitution.
It's the same with many europeans having trouble with the idea that you can't safely drink tap water, have no public health guarantee and education isn't equal for everyone in the US ;-)
On a more serious note; stuff like the US amendments is often already included in the base law (the generic name for constitution) as they are much older than the US. The times where the base law was somewhat more moldable (like before widespread literacy) made that possible. For example, freedom of religion is pretty universal, except in countries where a ruling religion exists which inherently to the religion bans all others. Arms are usually removed from base laws a few hundred years in as they directly conflict with the laws that ban killing people.
