Immigration politics in both places has gotten very ugly, but it strikes me that in Europe it may be worse than in the US. We may be about to see the fall from power this weekend of Angela Merkel as Chancellor of Germany over the issue of immigration, with her having been for some time the leading political figure in Europe supporting more moderate policies towards immigrants, even as she has had to retreat more recently from her earlier opening to a million migrants from the war in Syria (and also some others in the region). If so, this will follow the coming to power of anti-immigrant Sebastian Kurz in neighboring Austria, who is actively working with Merkel's anti-immigrant critics to overthrow her, not to mention the strong reelection of Victor Orban in Hungary and the assertion of anti-immigrant policy in Italy with newly installed Interior Minister, Matteo Salvini. All this follows anti-immigrant forces coming to power in Poland and some smaller other nations, as well as the anti-immigrant Brexit vote in UK two years ago. Only in Spain have we seen the recent installation of a friendlier-to-migrants Socialist government in Spain, which is however preoccupied with the Catalonian separatism issue.
Of course in the US we have seen Trump elected on a strong anti-immigrant wave, and he has just had a SCOTUS decision supporting his power to enact travel bans. OTOH, his recent awful policies towards asylum seekers on the Mexican border have led to a backlash that has forced him to back off somewhat, although what his actual policy there is currently somewhat unclear. He remains strongly anti-immigrant, but he did not win the popular vote, and he remains highly unpopular in the polls. His base clearly loves this stuff, but it does not seem to be selling with fervor or effect that it is in quite a few European nations recently.
An important fact here is that in terms of surges of immigrants, whether legal or illegal, both places have had somewhat similar experiences. When Europe was facing the surge from Syria, the US was also facing a surge from Central America. While there has been a slight recent uptick in the US, the surge in Europe is over. Immigration rates have returned to the "normal" of before the Syrian war, but somehow the anti-immigrant fever seems if anything to be growing, with this further a bit of a mystery in that overall economic conditions are at least somewhat better as well than they were at the height of the immigrant surge. The only similarity to that in the US is that much of the anti-immigrant Trump base continues to believe that the US is being overrun by Mexican immigrants, whereas in fact net immigration of Mexicans has been essentially zero for the past decade. Those coming in now are from the troubled "Northern Triangle" of Guatemala, El Salvador, and Honduras mostly.
So it would seem that with roughly comparable conditions in terms of immigration numbers as well as economic conditions (although the US is arguably somewhat better off on that than many of the European nations), the anti-immigrant fervor in Europe seems stronger politically than it is in the US now. Some of this may reflect broader anti-Trump sentiments in the US, with most of the anti-immigrant neo-authoritarians in Europe not as unpopular as he is (with some of them just freshly in power so not long enough to have engendered much hostility). But I suspect that beyond all that there is the more basic difference that indeed the US is fundamentally a land of immigrants, even if for some "nativist" Trump type WASPs, those immigrant ancestors came some time ago. But we have long had people coming in from many different places with varying degrees of success at assimilating them.
This has not been the general history in Europe, where most nations have their identities strongly identified with a single dominant ethnic group with its language and usually religion as well. Small minority groups have long existed in most of these nations, and many have sub-regions with different groups dominant. But that is it, even in those sub-regions, there is usually a dominant group, and these groups simply have not had the experience Americans have had of more or less steady immigration and assimilation from a wide array of foreign nationals over a long time. The US is simply better at it, if far from ideal, given our history. And so, while strong anti-immigrant policies in places like Hungary may lead to strong reelections for those carrying them out, in the US the support for such policies is far weaker.