Unique is not a unique word, at least not in the sense of having no synonyms, since “singular” is a good replacement, and the phrase “one of a kind” will also do nicely.
But unique is one of the “battleground words,” one demarking the realm of prescriptive vs descriptive grammar, one whose use or misuse sets teeth on edge.
My ears hear the phrase “very unique” and I cringe a little, so my ears are pretty well in the prescriptive camp, at least on the “unique” question. That’s not all that surprising, since I use words and grammar professionally and besides, as I think I’ve explained before, I have a long history of intellectual snobbery.
Nevertheless, I’m not committed to my prejudices. The singular “they” offends my ear (“If someone has an illness, they should see a doctor”), but I recognize that it is neither ambiguous, nor is it of recent origin (see Chaucer, for example). So in writing I’ll usually try to finesse it, as any battleground usage can interrupt the flow for some people, and flow control is important. Clarity and communication are the goal; grammar is merely the means to an end.
In the case of “unique” however, there are some other things going on. It’s interesting, for example, that adverbs that are usually intensifiers, such as “very” or “highly” actually weaken the usage of “unique.” “Very unique” becomes equivalent to “very unusual” which is less uncommon than something that is unique.
There is also a deeper philosophical argument here. It is the nature of a singular event that it should not really be comparable to other events, otherwise its singular nature is threatened. But we do not live in the world of Platonic Ideals, nor a Thomist world where each Angel is a separate species.
My own take on the matter originated from the contemplation of Cabbage Patch Dolls. (Hey, you take your insights where they happen). Originally, the Cabbage Patch Dolls were soft sculptures, similar in some ways to my wife Amy’s Bat-a-Brats and Sock-Hoppies. So each head was a “unique” craft creation. But when the Cabbage Patch Dolls went mass market, the design was changed for ease of manufacture, and their heads became stamped plastic. However, each Cabbage Patch Doll has a pattern of freckles on its face, and the freckles are varied for each doll, so No Two Are Alike.
So each Cabbage Patch Doll is unique. They’re just not very unique, are they?