no it's not.
National identity is defined by how integrated you are. Are you part of society and most importantly: Do you speak the local language?
Being born in let's say Germany doesn't make you German. Speaking German makes you German. In my experience this is actually true for a lot of countries.
Where you're born isn't as important as many on the right and on the left like to claim. It's all about integration + contributing to society (and to some extent not leeching on the welfare system). To most people it's just a simple matter of fairness.
Here are a few examples:
Have you ever heard anyone in Europe complain about migrants from Asia? or from rich Arab countries? Why not?
Have you ever hear anyone complain about eastern Europeans and gypsies? Why would anyone complain about them? they are all white Europeans, right?
This flies directly against the experiences as a non-White person - you get the question "no where are you
really from?", or the "wow you speak so great German", or you are labeled as a "second/third/fourth/fifth/sixth generation immigrant", you don't count as a "real German", etc.. All because of your skin color. And that's just the diet racism. Then you have the full-blown racists that have become the norm in media and news papers and politicians and on the street who call you racial slurs or tell you to leave the country or spit on you or threaten you. Even if you have a job and are born in the country, you are met with these things in the public and on the street.
Your example with "gypsies" and Eastern-Europeans doesn't hold water either - both those 'groups' (if we can even group them like that) are highly racialized and Othered by virtue of the way they look, the way they talk, the way dress, and so on.
No, I'd say European "racism" is first and foremost about a perception of unfairness. There is a strong mentality of "what we have today we accomplished all by ourselves after the war" present. With this kind of mindset how would you react if somebody comes in and expects a free ride (even if that's only perceived)? In this aspect, I fear Europeans are way closer to Americans than some would like to admit. The difference is, that Europeans are way more worried about this because they know the benefits a social security net brings and wouldn't want to lose it. I guess that part is harder to understand if you live across the pond.
And yet... you have to kill the net to protect the net, at least that's the logic some right wing parties seem to apply.
But then this is merely an issue of (false) perception and stigmatization of people coming from other countries. They work too, they pay their taxes, they do the same shit as anyone else, even being unemployed or have to go on welfare sometimes. Why should they not receive the same benefits as other people who are also working or who are unemployed?
There is also a false perception in that thinking that what Europeans have was earned through surviving the wars, when in fact that completely overlooks the riches and wealth that the countries have accumulated through colonization and Empire.