Words are important in activism. It's not just about saying the right things, or explaining things in the right way, it's also having the right words to describe what it is you're talking about. That's why I made the (probably futile) effort to coin the term gnostivism. Words can be harmful but they can also be helpful in mitigating harm. Identity labeling is considered by some to do more harm than good, and I really disagree with that. So I'm going to try to explain why I see it as beneficial, at least for right now.
The main argument against identity labeling is that it only serves to divide people. There is a certain notion amongst particular groups of progressives that by eliminating all distinctions between people we'll be more likely to treat each other equally as human beings. This is a fairly positive idea all by itself, and isn't a bad thing to want to work toward, depending on how one does that work. I would say that, in fact, I do believe that in an ideal world identity labeling will be de-emphasized and people would be allowed to just be people, in whatever way they happen to express that. However, jumping to the elimination of labels in any situation short of an ideal world can lead to a lot of problematic responses to real world issues.
Imagine you've been having a feeling. It's really affecting you and has been for a while. You have trouble sleeping, it's making you really depressed, you feel alienated, and you're getting desperate about it. You go to a friend that you trust to try to describe it, but it's vague so you aren't quite sure how. You tell your friend that you have trouble thinking about parts of your body as belonging to you. You don't associate with the way the people around you are acting. You can't understand the appeal of sex at all and you think there might be something wrong with you. Now, we don't live in a perfect society. In the real world there's a good chance this will end with one less friend and no help given, but let's argue that this isn't the case and your friend really wants to help. So how does this break down with and without identity labeling?
A policy of "just treat people like people" often discourages one from actually informing themselves about the needs of different identity groups, because that information is irrelevant to the goal of being kind to everyone. Not to say that this is the case for everyone that believes in label elimination, but this is often the interpretation of the philosophy for those who are members of mostly dominant groups. So in this instance your friend can assure you that you're a good person, tell you there's nothing wrong with you, and maybe act as a general sort of comfort. It's a nice reassurance to have but doesn't do anything concrete to address your discomfort. You're pretty likely to keep feeling depressed, and if people notice that you aren't acting like most of the people around you are you'll find yourself open to harassment and harm. There's no bad intent held by your friend, they just don't know any better than you do what's going on.
On the other hand, if your friend has seen, or heard of, or been exposed to an explicitly defined identity group outside of societal norms they might be able to point you towards something concrete. They may not be able to tell you that you are in fact an asexual or bigender, because the visibility of these particular groups is low, but they might be able to suggest that you're Transgender because they saw a news special about that community at some point. Suddenly there's something you can search, hundreds of books and articles you can read, a community that you can find online to talk to that may be able to lead you to your specific identity. You could find real ways to help yourself feel comfortable in your own skin, and have an understanding of your feelings. That's a lot of progress because of a word, and that word only came to you because of exposure to a group that uses it.
Having a word for something you're feeling makes it real and concrete. This is partially a function of the human mind, we remember connections better than ideas, and partially because that word means that whatever you're feeling has been felt by at least one other person. There is something incredibly reassuring about knowing that you're not alone in your feelings and an offer of a shoulder to lean on is hard pressed to compare.
In this instance those that oppose identity labeling would make the argument that the social construction of these identities in the first place is what leads to people feeling of displacement or dysphoria. They argue that ultimately oppression can only exist in a society that has individual identities which it can insert into a hierarchical system. On one level this is absolutely true, but it fails to take into account the fact that our society currently has hierarchical systems already in place using words connected to ideas to enforce them. Men are considered better than woman and man and woman have (usually incorrect) definitions attached to explain them. If we get rid of the words "man" and "woman" (not to mention the dozens of other gender possibilities) without disassembling the hierarchy first we still have a structure in which some people oppress others through male-coded derailing tactics, or the suppression of others through physical violence. The behaviors don't stop being taught and encouraged, but there is suddenly a lack of effective methods to identify the victims and allow them to organize.
Historically genocide and slavery in the United States continued even after religious colonization swept away the divisions that supposedly separated the humans from the heathens. Our modern understanding of race evolved out of equally evolving excuses to continue to oppress people of color even after they embraced the supposed olive branch offered by Christianity. Indian-Americans were long considered legally White by the US government due to Aryan ancestry, but they were still subject to oppression on the basis of their skin color. Queers have been bashed, imprisoned, and institutionalized by heteronormative power structures for over a century without ever identifying with a label or even necessarily having one. It's only because of the recognition between individuals that they share a common identity that these groups have been able to push back against these structures to begin with, and that fight is still a long way from done.
I would posit that when you claim to be racially color-blind or assert that pronouns don't really matter, so why not use "he" instead of "they," that this does more to disrupt the support systems of the people you're talking about than it does to end their suffering. The ideal of a fully unified society is just that, an ideal. Living as if the ideal is here now is a privilege that only those who aren't already suffering can afford. For the rest of us that ideal will take a lot of work to achieve. Whether you're with us or not you shouldn't be trying to take our labels away from us, because we don't just use them to label our boxes. We use them to label our flags as we march, fight, and demand that unification and equality.