Why I'm wary of labels

In various conversations over the years, I've noted my wariness of labels. While that trait isn't unique or even especially novel, it usually comes up against steadfast resistance. Part of that is just down to context. But people love their labels and insist on them. Labels are convenient. Now, I'm also someone who is self-admittedly obsessed with classification. So this might seem a bit odd. After all, wouldn't we use labels to classify things? Yes, but also, that's not really the problem that worries me. So here's something I just brainstormed that might help...

Suppose that A makes a certain claim.
Suppose that B makes another claim.
Suppose that those two claims are mutually incompatible.
Suppose that you might take issue with other features of one or both claims.
Note that A and B are both letters of the alphabet and that this classification can be used to group them together.
Finally, suppose that you use all of that to dismiss C, as C is also a letter of the alphabet and therefore is assigned a connection to both the claim put forth by A and the claim put forth by B.

That makes it pretty clear, right? You shouldn't dismiss C, poor C, because of something that A said, nor because of something that B said. Doing both simultaneously doesn't make it better. And yet, all too often, it seems like the use of demographic labels in discourse is a smokescreen to hide "straw man" arguments. Careful observation has led me to the belief that this isn't the goal of using labels. People want labels because it helps them compartmentalize, and compartmentalization is (most of the time) a useful cognitive tool. I see the appeal, but I also see it as dangerous. So that's why I'm wary.