I originally wrote this more than ten years ago, mostly to poke back at people I thought were being kind of rude. (For many years, the tone of this was a little snotty in response.) I occasionally get email about it, mostly "right on", occasionally "you're a moron". Generally I think it's a waste of time and happiness to get snippy about someone's grammar.
There's nothing wrong with "very unique". Considered in its entirety, everything is unique. Either that makes uniqueness a trivial and uninteresting property, not worth mentioning, or that's not what uniqueness is about.
If you were to consider every aspect of any two things, they would differ in at least one. Two physical objects always differ at least a little bit, in location if nothing else. Two abstract ideas are different by definition, else they're one idea, not two.
When people say something is unique, they're not just saying that there's nothing else identical to it; they're saying there's nothing else like it. But what does it mean for one thing to be like another? It means that they are similar in some important ways. The more ways (or the more important the ways in which) they're similar, the more similar they are. 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 hippopotamus.
The uniqueness of something is more apparent and more interesting when it is unique in ways that most things (or most similar things) are not. 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. A person made of marmalade would certainly be unique, but if that person were also six meters tall, they'd probably be unique in three unusual ways! (In addition to the obvious two, I'm guessing they might be the only six meter pile of marmalade.)
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. It's not good style by many people's standards, but it's not without meaning and it's not unclear.
All that said, it's probably best to avoid using the phrase, simply because many people will complain about it, and you'll end up having an argument about grammar instead of talking about that other interesting thing.