A wooden chair is like a wooden table, but it's usually more like another wooden chair. When someone says something is unique, they mean that nothing else is like it in a bunch of ways they think are interesting. For example, The chair I'm sitting in is unique in that I'm not sitting in any other chair at this moment, but that's not very interesting. I'd be more likely to point out the uniqueness of a chair that was made of titanium and had a fuzzy pink cushion in the shape of a wombat. All people are unique, but a person made of marmalade would be unique in a way that most people are not, material composition.
Uniqueness is also more interesting when there are a number of such interesting feature subsets.
When people say that something is very unique, they're saying that it's unique in an unusual way, or that it's unique in an unusual number of interesting ways. I think it's a fine thing to say, except around people who think they know better.