I identify as a Korean-American. I do so simply because I am an American citizen, and I believe that my ancestry goes back through Korea for quite some time.
I would imagine that this is typically how people identify with respect to their ethnicity. However, there are other ways of thinking about this.
Suppose you have an individual whose biological parents have ancestries in Korea, but who was adopted and raised by a white American couple. Suppose also that you have an individual whose biological parents have ancestries in Europe, but who was adopted by a Korean couple in Korea. Who would you say was more "Korean?" Do both have equal claims at identifying as Korean?
When it comes to identifying as an ethnicity or nationality, genealogy matters, of course. But if all you have is genealogy, then it seems that your identification is shallow. Substantive ethnic identification involves more than just being born into a certain family tree. One has to be deeply embedded in the culture of said ethnicity.
There are at least three ways of engaging in culture.
1. Consumption and production of commodities
This includes stuff like making and and eating an ethnicity's food, wearing ethnic clothing, consuming entertainment produced by that ethnicity, etc.
2. Compliance with norms, customs, and practices
This is stuff like dining etiquette, rules of the road, entertaining guests, participating in religious and social rituals like marriages and funerals, etc.
This is self-explanatory. This is attaining fluency in the language associated with the ethnicity in question.
I listed the above in what I take to be the order of shallowest forms of engagement to the most substantive. It seems clear to me that the most substantive way of really engaging in a culture is to master its language. In fact, you won't even be able to do the other two very well without fluency in the ethnicity's language. Language allows one to understand an ethnicity's perspective and worldview. I hold that without fluency in the language, ethnic identification remains shallow.
This means that although I identify as Korean-American, my ethnic identification as Korean is pretty shallow. I can speak some Korean, but my language skill is pretty rudimentary. I eat Korean food, know what to do during Korean holidays, know how to interact with my elders, etc. But all this is merely rote behavior. I don't have the ability to read Korean at a high level, so I can't read Korean commentary on Confucius. Nor can I read the famous Korean poets and storytellers. My linguistic limitations prevent me from really taking on the Korean mindset, and thus keeps me from really identifying with Korean culture.