Why are Serbs always pointing out their nationality?
1) It is a immigrant thing. When you feel that you are being identified as the "other" you get the urge to preemptively "come out of the closet" as an immigrant in a show of good will towards your native hosts and try to fit in. It's a way of saying "Hey guys, I know I am this scary foreigner, but really I'm just like you so can we be friends?" I've experienced this with people of many other nationalities and ethnicities where the conversation will inevitably turn into them telling me about their country of origin.. which I don't mind since I enjoy learning about different cultures.
2) Maybe you encounter a disproportionate number of Serbian people in your daily life so point #1 is magnified, or perhaps because of your own background or personal reasons you are overly sensitive to the idea of Serbian identity.
3) Serbs do tend to be sensitive about their ethnic identity because many of us feel that we have been misrepresented and not entirely fairly villainized in the media, while at the same time also being victims of long standing ethnic hatred in our native countries, so insecurity borne out of that treatment motivates an exaggerated sense of nationalism in some of us.
Hopefully one of those, or a combination of them, explains why "Serbs are always pointing out their nationality".
If you're from a place that doesn't get a lot of international press, it's natural to feel pride when a person from that place is met with acclaim. My high school teachers probably still talk about the fact that most of them also taught a writer for The Simpsons.
They're all interrelated though, so it gets complicated rather quickly. But yes: Serbs are proud to be Serbs.