* US Americans are generally very sensitive to the issue and try to be PC as much as possible.
* Europeans are rather insensitive to it, which shows in two ways: A lot of people seem to be more color-blind, which IMO is the upside, and a smaller but very visible bunch of people are more openly racist.
* This also applies to public discussions - the skin color of immigrants or inhabitants is rarely mentioned, people and the media rather just differentiate between natives, people with immigration background and immigrants - here in Switzerland maybe with special mentioning of Germans, French and Italians, as those three originating countries tend to be the most prevalent. Thus, it seems much less a race-, than a nationality issue.
* On the topic of numbers, I gotta say I think it depends. Here in Switzerland roughtly 30% of inhabitants don't hold Swiss nationality, tendency rising rather quickly. Of those, roughly half come from neighboring countries (i.e. culturally close), the other half is mostly from Ex-Yugoslavia, Middle East and Northern/Central Africa. Switzerland today seems much more like a melting pot than it was a decade ago, which is something I actually appreciate (I married an Asian immigrant myself and my son has two nationalities, which is actually a majority phenomenon in the whole greater Zurich area now).
> differentiate between natives, people with immigration background and immigrants
Yes, in a racist/othering manner. Sure, in the US you might hear about someone being an immigrant or being born to an immigrant family. But this is almost universally mentioned as a positive reflecting ones Americanness. With illegal immigrants being the only notable exception. In Europe it's often the other way around.
This "it's not about race, it's about nationality/whatever" gambit is exactly how Europeans fool themselves into believing there isn't a problem. It's not about the word, it's about treating people badly for who they are.
Although saying that, I do have a picture on how heavy regulation could work better - have more stuff selected blindly. And I do mean blind - no names, no voices, instead you get assigned a case ID (from government?) and you communicate through secure chat, until an offer is either accepted or rejected from any party. That could at least meet discrimination at the initial selection process - of course you still have adverse networking effects leading to lots of jobs not being offered public anymore and the subsequent workplace discrimination issues - these are harder nuts to crack. But the way e.g. job applications in the US are handled is an incomplete shield, it should be either all-in blind or just open to 'cultural fit' questions from start IMO.
Being PC doesn't mean not being racist. They will keep their thoughts to themselves and will not let you know directly. But behind your back, oh boy...
I hear people talking trash about 'brown' refugees on a daily basis here in Paris. I've probably heard something similar once or twice in my 8 years in the US, mostly coming from mentally unstable people like the ones you see in the streets in San Francisco.
TBH UK is a lot more sensitive than the mainland Europe (there's a lot more immigrants in London too).
Anecdotal, of course, but that has consistently been my impression.
Of course, the U.S. has other issues (an increasingly violent police force as well as income inequality) which makes the effects that stem from racial discrimination worse than they would elsewhere, although that's just my take.