|Twilight Sparkle is only obliquely connected to this post.|
What do we really mean when we say that? I can think of three major motivations for expressing that sentiment, or some variation of it: concern about possible misclassification, conflation of group membership with self-identity, or general wariness of labels.
Whatever our stance on classifying people, just about everyone dislikes being misidentified. Of course, strangers or casual acquaintances often have no way of knowing certain things about us. For example, people often assume I am gay because I have a male partner when I am, in fact, bisexual (the phenomenon of bisexual erasure may also affect this in some instances, but I am referring to people with no access to any information about my sexuality other than my present relationship).
Conflating group membership and self-identity is similar, but more subtle. Group membership often matches self-identity, and many people consider them one and the same. If there is a discrepency between them, we might feel as though we have been misclassified, even if that is not the case. For example, I am a citizen and resident of the United States, which makes me a member of a group called 'Americans'. However, I did not grow up in the USA, and do not identify strongly with the culture thereof. I do not really think of myself as an 'American', even though I am one by most relevant definitions.
Some people just do not like labels in general, even accurate ones that harmonize their their senses of self. I suspect there are many psychological reasons for this. People who grow up enduring social torment as outsiders--'weirdos', 'nerds', etc.--might reject any kind of label. Some argue that labeling is a kind of objectification, a way to reduce people to stereotypes. I do not really think we need an example here, but for the sake of symmetry, I match all of the common criteria for the label 'brony'.
The nature of the human cognitive process inclines us to classify everything, including people. That does not make it good or bad in itself, just nearly inevitable--at least given our present level of technological sophistication. We almost certainly all do it, whether we realize it or not. The biggest (and perhaps only) problem with a sentiment like "I don't like being labeled" is that it sets standards for other people that we ourselves probably cannot meet.
I think of labeling not as reducing someone to stereotypes, but linking someone to concepts that are helpful in thinking or communicating about them. These include physical descriptors, location (both in cyberspace and meatspace), interests, cultural/subcultural identification, connection to other acquaintances, and so on. Maybe I only see it that way because I rarely feel uncomfortable with being labeled myself (with the exception of misclassification, which I am working on).
Or maybe I am overthinking this, just a little.