Why are people so insecure that they are afraid of being labeled?
Because there are umpteen historical instances of people actually being not only discriminated against but also killed/murdered because they have been labeled into one kind of category or another, often arbitrarily with little basis in fact except prejudices. The appreciation of a particular label can and does change over time. And once someone is labeled, just try and erase the memory of being labeled.
Who would want to be labeled as a witch in 16th-17th century Europe? Or as a gay in parts of the present Mid East or certain African countries? Who would want to be labeled a pinko in 1950s USA, or a Trotskyite in 1930s USSR, or a capitalist running dog in Mao's China, or a Jew in Nazi Germany?
Labels are often assigned arbitrarily on flimsy basis and little evidence. And even if they might be considered sociologically relevant, labeling someone might very well have very adverse effects on real people with real consequences.
And since the way people look at a certain label can change at the drop of a at, it is not surprising that people don't always like to be labeled or pigeonholed.