Interesting...well maybe this is a case where looking at it from an American perspective is causing me to misread things a bit. When I was in college, I did have a few friends who were foreign students from various Latin American countries and, unsurprisingly I guess, tended to be from very well-off backgrounds. And I noticed exactly what you're saying about their self-identification--that they tended to emphasize that they were 'pure European' or things to that effect. I always found that puzzling, and because I got the impression (perhaps wrongly) that they meant "European as opposed to 'Hispanic' " rather than, say, "European as opposed to Peruvian/Salvadorean/etc.," I just assumed this was primarily a racial comment, that it was more or less a way of insisting "I am white" and that this was important to them for some reason. But perhaps it was meant strictly with reference to social hierarchies within their own countries, and nowhere else?
It's a rather complicated issue. Latin American people, except perhaps native Latin Americans (meant as a "Native American" counterpart), in general don't have a clear identification with their countries further than pulling out flags when their national football teams play. Similarly to the US, these countries have grown on immigration, mainly from Europe, but differently from the US these immigrants, for a number of reasons have not set roots in these countries in the way immigrants to the US did. There was always a nostalgia in them, a desire to return to the homeland, possibly because these countries, while giving them opportunities never made them feel at home and never gave them anything to be proud of, something which of course they, as part of the society, didn't contribute much to, but naturally would never recognise.
This results in a lack of true identity and a desire to belong. In some way white Latin Americans are like adolescents: they don't know exactly who they are and want to be part of the "cool" group, which to them is not home but rather the first world. But then, while they revel in their first world fantasy (brand clothes, technology, etc.), they know, though they'd rather not, that they are still Peruvian, Brazilian, Argentine, etc. and that's how people in the first world see them. Maybe to Europeans and Americans this fact is not necessarily pejorative, but to the great majority of white Latin Americans it is, as a result of their own perception of their homeland. And this leads to the resentment part: "we want to be like them but they, with their policies don't let us".
About asserting their "whiteness", there is something to it because there is a strong racism in these countries, especially towards natives, probably because they remind them where they really come from. But then of course everybody, with typical first world political correctness will be utterly horrified at the mere idea of discrimination, though in the safety of their trusted circles will be nonchalantly deprecating the "negros de mierda" (literal translation: "fucking niggers").