I think /u/Alorha's answer currently does it best. There's not much to it. People obviously know they're white, but it's not a huge part of identity for most people. The Whiteness Project focuses on the same question that you're asking, OP, and personally I think the answers are rather interesting. It's something non-whites in the US (that's what my experience is limited to) don't seem to understand. When you are treated differently based on your race, it's easy to see how your experience differs from the people who are treated as "default human." From what I can tell, it often becomes a significant part of your identity because it alters your life experience so much. White people don't have the same unity because they don't have to.

The whole privilege issue also leaves a bitter taste in the mouths of many white people. Many people seem to take the "you're white so you experience privilege" as their life being easy, experiencing little to no difficulty in any form; in reality it's just that it's easier.

Personally, it's annoying to me that so many white people I've met consider it weird to be proud of your heritage. To me it doesn't make sense to have pride in being white, but in your heritage I can understand. I can share in the frustration many white people when all white people are occasionally said to be the same: a hateful, WASPy type that is resentful of all minorities. Being that this same group has hated me and my family for a long time (Family came to the US after the Easter Rising and experienced prejudice working as maids for the owners of steel mills; I grew up in a Catholic school where immigrants outnumbered children born in the US) it bothers me that I am now lumped in with this group, but I accept it because it's not worth fighting. Yes, I still benefit form white privilege but sometimes there are situations that make it difficult to take it sitting, when I am told that my ancestors, despite being slaves as well, were the ones that brutally enslaved others. However, you just have to take it if you believe in the point being made, because there's no sense in making an issue of race relations into a discussion about white people.

In short, you do experience white privilege, obviously. There's not a sense of unity or a feeling that being white is part of your identity. There can also be situations that make it difficult to be lumped in with other white people (especially during protests related to race), but the best solution is really just to take the hit if you believe in the point being made.