People who are half-white, half-black identify as black more often than half-Asian, half-white or half-Latino, half-white people identify as fully Asian or fully Latino. I believe there are several reasons why this occurs.
One is because of America's history. When slaves first came to America in the 17th century, there was a lot of rape between masters and slaves which caused women to give birth to children who were half-white and half-black. However, because they were the child of a black mom, they were still considered to be a slave. The one-drop rule only intensified this reality. The rule stated that if you had any black in you than you were 100% black. There are some public figures during the time of slavery who were mixed with white, but were still slaves and therefore considered black. Booker T. Washington is one example.
However, there are public figures in the present who society still identifies as black even though they have parents of different races. This is caused by that fact that many of these people look similar to people who are fully black. People like me. I identify as just black, but aesthetically I am a lighter shade of brown I have green eyes as the result of two African American parents with mixed ancestry primarily from the 19th century. There are many other people like this, such as Tyra Banks, Rihanna and Chris Brown. But, because you can have two black parents and be a lighter shade, society views people with a white parent as fully black as well. Halle Berry, Barack Obama, Alicia Keys, and Mariah Carey are a few examples. Halle Berry has one white parent and one black parent, but she identifies as just black because of the one drop rule and even advocated that her daughter be only black too, even though she has mixed-race parents as well. Mariah Carey considers herself fully black in this country because of the one drop rule, even though her mom is white and her dad is half-Latino and half black. Here is a video of her on the George Lopez Show talking about her race:
So how should these celebrities identify themselves? If you have a higher-than-normal mixed ancestry, but your parents and grandparents are black, then you normally would identify as black, but what if you have parents who are different races? This can be complicated. Since so many African-Americans in the U.S. have a lighter tone, it is very plausible that, if you are interracial, you are seen as black to the average person on the street. Obama could have easily run into racism similar to what the average black person experiences. However, race is not only an external idea, but it is internal as well. There is the idea of community.
Obama grew up in Hawaii, but then moved to the south-side of Chicago where he attended a black church and was around a lot more black people, marrying a black wife while there. So he probably felt blacker in Chicago than in Hawaii. However, race is affected by the members in one's family as well. Because Obama has one white parent, it might be easier for him to associate himself with white people with so many extended family members who are white. So, it would be unfair to ignore this privilege and put his view of the world as synonymous to the way someone with two African-American parents views the world and the people within it. Interracial people might also get a slightly better understanding of where they came from. For example, Obama is 50% Kenyan and 50% White and knows which countries his white-American ancestors descended from. So he has the privilege of knowing his heritage.
Because of slavery, most African-American folks cannot trace their history back to what countries they came from before the U.S. Those with a more substantial mixed heritage, like me, do not fully understand where the mixture came from or at what time. However, regardless of ancestry, because all of my living relatives are black, I identify as black. But, if you have parents of different races, you might struggle with how to identify racially. Overall, my experience with race is radically different than someone who has parents of two different races, so society should view interracial people and people of only one race separately.
How should people who identify as two races go about their identity? Quite frankly, I do not have a solid answer for that, because one size never fits all. But, I feel that is important to identify with both sides. One way to go about it is to identify not as two halves, but as two wholes. If you are interracial, then often you are taught as a little kid that you are in the middle so you do not fit in with anyone. This rhetoric of being only half of both things can be problematic. Instead, it might be best to celebrate both sides of yourself as much as you want, because both of them fully encompass you.