I don't think you can say [people tend to emphasize their non-white ancestry rather than their white ancestry] much anymore. Living in California it's pretty obvious people of white/asian and white/latino background don't fall into the same tendencies as people of black/white heritage. They don't deny non-white heritage, but since they look white and are treated as white, unless they are SJW types who are faking it, they pretty much assume they are white.If by adopting an assumption of whiteness he means they don't adopt conspicuously underclass behaviors mostly associated with blacks like mixed black/white and lower class whites often do, I get it.
If, however, the contention is increasingly the case in formal or professional settings, I'm not so sure I do. Here are racial self-identifications among GSS survey participants of Mexican ancestry, Asian ancestry, and Native American ancestry, respectively, by year. The question includes just three possible responses: White, Black, or Other. The middle answer is mostly negligible across the groups considered and is therefore subsequently excluded.
For those of Mexican ancestry living in the US:
The transition away from white and towards other self-identification is also apparent in the US Census, where 42% of those who identify ethnically as Hispanic choose the option “some other race”, a racial categorization that is almost exclusively used by those who identify as Hispanic.
For Native Americans:
The trends are less pronounced for these two groups but they move in the same direction; in any case the patterns here don't appear to be towards greater association with whiteness.
As has been previously noted, there has been a marked scramble away from Americans identifying as ancestrally English over the last several decades, resulting in a reported decrease so drastic that it might suggest, if not merely the result shifting preferences, that secret but deliberate ethnic cleansing was taking place in the US.
GSS variables used: RACE(1,3), ETHNIC(5,16,20,31,40)(17)(30), YEAR