It's easy, being white, to not to recognize culture you do have. It's easy to put on a role or culture -- anyone can -- but the culture around you ignores your "background" culture and sees it as an assumed role; this role or culture becomes your new descriptive label.
For example: in college I could experiment with what my life role was going to be. I tried on "a nerd" and "an athlete" and "a hippie" and even "a gamer", among others. In each of these communities, I fit in to the degree that I had the skills and mental attitudes that that community rewarded. I was called by those names by my friends (and enemies) and peers, who would use them to describe me and distinguish me from others who might superficially be similar.
But a black man in the same school would have had the same opportunities to do all the same activities. Let's suppose he had. He would have been described by the same people as "a black nerd" or "a black athlete", "a black hippie" or even "a black gamer". (perhaps they would have said "Black" instead, or "African-American", but the point is the same.)
In short, a white American Anglophone like me can be "that guy who's into [e.g.] Chinese culture", but a Chinese national and native-speaker of Mandarin would be "that Chinese guy who's into Phish". It's this freedom to put on roles and not have your underlying racial group be attached to them that is an important aspect of privilege. Whiteness is invisible, and even its invisibility is difficult to see.
This is not to say that you shouldn't use the word "black" [Black, African-American] -- rather, that we should recognize whiteness (or European-descent-ness). I try to describe and distinguish people who are white by saying "white" instead of just perpetuating the default assumption. Compare these two sentences:
- "he's about 5'9", long hair, kinda scruffy"
- "he's a white guy, about 5'9", long hair, kinda scruffy"