Race is real. It is a real social construct. The race(s) that people perceive you as being a part of have a real effect on the others perceive you, what they think of you, and how they treat you. Your own perception of your own racial identity affects you as well. Racial identity is closely tied with community, family, and heritage. And the construction of one's racial identity is extremely complex -- not easily reducible to a handful of simple, universal guidelines.
There's been a lot of discussion about the (former) professor in Montana who chose to identify as black even though she was white! Personally, I think that what she did was extremely misguided and indeed dishonest. But I think it is also misguided to try to come up with a simple set of guidelines about who is allowed to identify as which race just so we can have some general, objective standard for criticizing her. (I don't want to pick on this one article, but here's an example, for reference).
For one thing, I don't think it's useful to insist on having a theoretical framework that works perfectly for both racial identity and gender identity. That's like insisting that your framework for gay marriage must also fit poly-marriage: there are very clear and real parallels, but there's a sufficient difference in complexity that you can't treat them as interchangeable.
What qualifies as a race anyway? Some choices include white, black, Asian, Pacific islander, Native American, hispanic, Jewish, Muslim -- I've even seen Brazilian and Hawaiian treated as "races". I'm not trying to say "Ha, ha, how silly to consider these 'races' when we all know a 'real' race is XYZ..." -- quite the opposite. These can all be categories that affect how others perceive you and treat you and how you think of yourself. But which categories are perceived as real categories depends very strongly on the culture and society you're in. You probably have multiple racial/ethnic identities. Your racial identity may change depending on your own experiences and on which group(s) you're currently interacting with.
But this Montana lady is white, and hence she was pretending to have faced discrimination and disadvantages she never faced! True, and very problematic. But I would be wary of using this example to define general guidelines. There are plenty of people with mixed heritage who look so white that no one would ever spontaneously guess they're something else! They don't face quite the same discrimination as others in their family and community. I read lots of blogs and articles by folks in this category (in addition to knowing some personally), and I've consistently seen the message that such people understand that they experience privileges that their fellows don't. Yet, if their racial/ethnic heritage is important to them, it would be totally inappropriate to tell them they have no business identifying as XYZ if they look white.
But this lady in Montana intentionally made herself look black and lied about her black heritage to become a leader in the NAACP! And that was dishonest (not to mention deeply weird, considering being white wouldn't have disqualified her for the position). It was unethical. But that doesn't mean we should conclude that you can't change your racial/ethnic identity. Nor should we conclude that you can't go from a more privileged identity to a more marginalized one. Nor that your racial/ethnic identity has to derive from your ancestral heritage. Think about the categories on my list above. Some people integrate into a new group to the point where they identify with the new group completely (and are seen by outsiders as part of that group) -- and there's nothing inherently false or dishonest about such a conversion.
I would not judge those who don't want to identify with some part of their heritage, regardless of whether they're rejecting a more privileged or a more marginalized identity. I think we should respect people's own decisions about which parts of their heritage (and which new communities) they identify with and which they don't.
It doesn't make sense to suggest that a person has one true race that can be objectively determined and defined for them.
As far as genetics are concerned, I assume I don't have to explain to my readers that all attempts to use genes to canonically divide the human species into races has failed completely. Yes, the color of your skin, the texture of your hair, and the shape of your facial features all depend strongly on your genes. But the handful of superficial features that are used culturally as cues to determine someone's race are essentially worthless for dividing people into any kind of meaningful genetic groupings. The most distantly-related human populations in the world -- aboriginal Australians vs. some populations in Africa -- get simply grouped as "black people" when we use these superficial traits. In the meantime, some populations of "white people" and "black people" are much closer to each other genetically than either one of them is to the populations mentioned above. The experience of race is real -- members any of the above categories of "black people" would probably have similar experiences if they moved to a majority-white country. But the genetic basis for grouping them? Not so much.
My son's two best friends are an interesting case in point. His one friend is a Muslim from North Africa, and the other is... well... his dad is Mexican, and his mom is mixed further: her dad is from Haiti and her mom is French Canadian. So "racially" he's half Mexican (is that even a "race"? -- he doesn't speak any Spanish...), and he's a quarter "black", and, I guess, a quarter "white". I've heard that the French Canadians, like the Mexicans, have a high proportion of Native American heritage, but, from a superficial-racial-characteristics point-of-view, they're basically indistinguishable from any other white people. Um, except the ones that "look black"...? Anyway, the funny thing is that from a superficial-racial-characteristics point-of-view, the two friends could easily pass for being of the same ethnicity as each other, despite the fact that their backgrounds have nothing to do with one another. These superficial characteristics affect our lives so much, but it's sad that we place so much weight on them, considering that they are so completely arbitrary.
Now, I will add one caveat that I think there exist cases where it makes sense to question someone's choice of racial identity. The Montana professor, for example -- she was deceptively implying a set of experiences she didn't really have. For another, I'd question folks who use a trivial ancestral connection to harm others, eg. folks who claim that they had some Native American ancestor, hence their opinion on some team name not being racist should be taken seriously. But outside of such obviously problematic cases, I think we should err on the side of allowing people to define their own racial identity based on their own experiences.