apushatayid, says who basically answered you. While the bottom line is that we are all fellow Yidden, there are things that categorizes us as individuals or groups. Sephardish and Ahskenaz, Chassidish and Yeshivish are examples that are NOT derogatory, they just describe the mehalach of an individual or group.
Of course, within each group there are subgroups and then individual characteristics. But Chasidish, Litvish, Yekkish, Sephardi, Yemenite, these are that labels LOOSELY but not negatively describe an individual. While it is true that that not everyone is correctly termed with the right label, the labels still have a definition that cannot be applied for everyone.
A man wearing a shtreimel on Shabbos, his first language is Yiddish, attended a Chassidishe Yeshiva, oops sorry no labels, okay learnt in Belz Yeshiva as a bochur, got married after seeing his chosen kallah after two beshows, learned two years in kollel and then went out to work can we call him Yeshivish?
There’s nothing wrong in saying he’s Chassidish. Of course, Chassidish doesn’t mean that everybody who is classified as Chassidish has the exact same lifestyle as the example that I mentioned above, but a label sort of gives some sort of idea of a person.
I’m Chassidish and proudly so. I see absolutely nothing wrong with being labeled a fellow Chassidishe Yid.