The Whiteness Project is an interactive investigation into how Americans who identify as white, or partially white, understand and experience their race

It's interesting to see some research into how white people perceive themselves, normally most of the research is in black / hispanic / asian communities.

As a white person myself, I expect the responses to be quite muddled, and for there to be very little consensus about what "being white" means. Unlike the other racial groups, most white people were never pushed into dedicated communities, and particularly in the Northeast and the Midwest, the time of their ancestor's immigration is so far back that we're likely to practice little, if any, of their original customs. I'm a partial second generation immigrant myself, and I only speak basic German at best and don't really feel all that connected to Germany, particularly as I've gotten older and encountered some of the starker societal differences between the US and Germany. If I ever adopt a kid, even though I would try to teach what I know, they would become further disconnected than I am, and so etc. Many other white people in the US have already undergone this for 10 or 15 generations, by which point it's incredibly diluted.

The other thing also is that since white people are generally the majority, anything that features white people is likely to be taken as more general than it would be if it were centered around other racial groups. Ed Edd n' Eddy for example, is a classic (or at least, heavily classically inspired) Canadian-American kid's cartoon, with a nearly all white cast, that takes place in a very stereotypical middle class small town / minor suburb. Yet lots of people watch it without really thinking about the race of the characters, just as a generalistic kid's cartoon, and it doesn't feature really any emphasis on racial stereotypes. But if you look at a show like Everybody Hates Chris, which was a reality show centered around a black family, it instantly stands out as a show produced around a minority, and the show is filled with references to things black folks might get that white or asian people wouldn't.

I lived in Japan for awhile, and there I experienced kind of the opposite effect. Any ordinary character in an anime was basically taken to be Japanese, and often there wouldn't be very much focus on what being Japanese meant. But if an American character is featured, such as Bandit Keith in Yu-Gi-Oh or Revy from Black Lagoon, it's always with an immediately stereotyped role - with the former, an obnoxious burly dude with an American flag bandanna, and the latter is a badass dual gun wielding chick raised on the streets of NYC. I'm sure that if there were a Japanese anime centered around an American family in Japan, it would also be inherently self-aware about what being a minority in Japan is like, while the same show centered around a Japanese family wouldn't.

TL;DR Just a really long winded way to say that I don't think there will be much consistency or a strong identity because the majority doesn't spend much time contemplating its differences vs the minority. Straight people don't really have much of a consistent idea of what "being straight" means, whereas LGBT people have our own subculture and common experiences centered around our sexuality / gender identity. I expect a very similar situation with racial groups too.