The wording suggest the opposite. Something that is one of a kind is but one of a category of many, if you look at each word non-idiomatically. Why, then, does "one of a kind" mean "unique?"
"One of a kind" does mean one of a category, as you say. However, it is one of a category of one, meaning the category only has one thing in it: that thing being referred to.
You also have to appreciate that this phrase is idiomatic. Taking it literally means that the person is one of however many that make up the "kind." It does not expressly say that the "kind" only has one member. Were the phrase more literal, or at least more exact, it might say, "the single of a singleness." Idioms do not make literal sense, though. That's what makes them idioms.
One of a kind could mean 'unique' because other less unique entities may not be one of their kind - with there being multiple entities of a given kind.
Think of it in terms of taxonomy of genus/species. You are a human but you are one that really stands out among your peers, a prime specimen, as your own type of human. You are one of a kind. Being one of many doesn't necessarily mean that you are not unique when everyone else just blurs together as the same thing. I think this is the essence of what the phrase means. I think it had meaning, but the dialect and slang has changed so much now over years and years it appears idiomatic now, but in reality it isn't.
"of a kind" definition from dictionary.com:
- of a kind, of the same class, nature, character, etc.: They are two of a kind.
So "Two of a kind" means "these two have the same characteristics as each other". It doesn't really say whether there exist other items that are from the same kind; there may be only those two or they may be another two thousand; we don't know.
The phrase "one of a kind" literally therefore means "this item has the same characteristics as itself", which makes no real sense. But it is not used literally, it is used figuratively to mean "no other items shares the same characteristics as this one".
It's almost a joke: "we can't say this item and any other item are two of a kind, so we'll call it one of a kind instead". Imagine a competition which is normally entered by teams of people, but one person enters on their own; people might joke that this person is a "team of one". It's very similar.
Of course, over time "one of a kind" became an idiomatic phrase that stood on its own, and no-one really thinks about whether it makes literal sense or not.