If something is unique, it's the only one around. You might think getting a nose ring would make you unique, until you see five other people on the street with the same piercing.
If someone calls you unique you know that you are very special. The reason is that the Latin prefix uni- means "one." (Technically speaking, it isn't possible to be "very unique" — you either are or you're not.) Think of the prefix in unicorn for a creature with one horn and unicycle for a cycle with one wheel.