I was born in Austria, and yet I do not identify as Austrian. My mother was a Russian Jew, which I consider myself, and I would identify myself as European before I identify as Austrian. Heck, I would prefer identifying as Sammarinese before I identify as Austrian although I only lived for a few months in San Marino as opposed to Austria for most of the rest of my life.
I do not have to celebrate or live the culture of a country in order to be a functional part of its society. It is none of your business what I refer to as myself culturally, how I celebrate my holidays, or if I eat pork or not.
There's a German city, I forgot its name. They have a lot of immigrants, but those immigrants are Japanese. They have Japanese schools, Japanese kindergartens, Japanese businesses, Japanese stores with Japanese products and Japanese price labels, where everyone talks Japanese. And there's no problems there whatsoever. You know why? Because everyone is tolerant of each other.
Saying that your co-worker identifying themselves as Turkish is a problem is, in fact, the real problem. So what if they identify as Turkish? Isn't that up to them?