I think it's obvious that what people mean is that "more unique" = "unique in more dimensions" or "the degree to which this differs from the norm is greater".

E.g. (2, 4, 7) (2, 4, 8) (2, 8, 4) (2, 4, 7) (1, 4, 8) (0, 9, 3) (987, 4, 7)

When asked what are the "most unique" sets in that list, you'd probably be acting deliberately obtuse if you chose anything but one of the last two.

I guess it's understandable but why not choose an appropriate word like "different" and keep "unique" a very strong word?

Well, you don't really get a choice. I'm just describing how the word is used, not suggesting an alternate meaning for it.

Math has long tradition of generalizing definitions to make sense for larger domains, when it's useful. See fractional powers, complex numbers, quaternions, and many more.

I can imagine defining uniqueness as a function returning real number from <0; 1> instead of a boolean value. For example:

    let U(p, x, X) be the uniqueness of property(function) p(x) for element x of set X 

    U(p, x, X) = 1.0 - (size of X')/(size of X\{x}), where
    X' = set of all elements x' of X such that p(x')=p(x) and x' != x

Property p of element x of set X is strictly unique when, and only when U(p, x, X) = 1

When it's useful? For example for speaking about minimizing collisions of hashes for given data.

Another way of thinking about it: uniqueness is 1-probability of uniformly randomly finding element with same value of p in X as x after removing x from X.

In a mathematical world you operate with abstract objects. In the real word - you need to abstract things; before that everything is unique; after that - well, depends on your abstraction. So unless you talk about mathematics, things can be more or less unique.

This is a excellent and subtle comment. You seem like someone with a tolerance for philosophical nit-picking. Please forgive me if I'm mistaken.

Instead of saying everything is unique we could simply say that there is nothing. A thing is itself an abstraction. The concrete world is without inherently distinct things. We must abstract things for "unique" to describe something at all. As you implied, this process is arbitrary. Every way in which you could abstract things implies a distinct notion of "uniqueness". To simply select one "uniqueness" (like mathematics) is arbitrary. But to consider every possible "uniqueness" equally is also arbitrary. Without prioritizing forms of "uniqueness" we can only construct a partially ordered set. So when you void a fixation on mathematics, things can be more, less or "incomparably" unique.

I suspect most pairs of things are incomparably unique. Further, I suspect most binary qualities are predominantly incomparable. I don't know that you should never say things like "more unique" but it might be fair to issue a warning in a prose linter. Any binary quality used as a continuum requires an arbitrary combination of it's distinct forms. If this isn't specified then it only has meaning for those who already know what it is.

Some philosophers, thinking especially of Graham Harman, have started reacting against the now sort of commonplace idea that "there are no things (or objects) in reality."

From a common sense perspective, it's obvious that there are things. Sure, you can point out the flux and decay of all entities, but still, this table here is a coherent thing even if it's made from parts in a temporary arrangement.

In some sense, philosophy itself is destroyed when you go down the path of denying objects, since philosophy crucially deals with concepts, and concepts are "thought objects."

Harman describes two modes of denying objects: undermining and overmining. Undermining is the tendency to say "really, this object is just a composition of these other particles," while overmining is the tendency to say "this object is just a modulation in a grand monistic entity."

Instead of that, he recommends an ontology of objects that's pretty interesting and fun to read about. He would, I think, agree that objects are unique in that they are (in programmer jargon) "pointer equal" to only themselves... and each real object, for that reason, has an infinity of potential that's never exhausted by any "arbitrary" perception of it... yet still, we perceive other objects not directly, but through aesthetic caricatures, and on that level you might have different degrees of uniqueness.

Thank you very much for this comment. I'm an armchair philosopher and I hadn't heard of Graham Harman. His notion of objects is beautiful. In one motion nihilism both compels me to accept my sins and deprives me of any path to salvation. Harman's objects capture the essential impetus of nihilism without ultimately voiding conception. In fact, they even capture the paradox of nihilism. The denial of objects necessarily implies an objective system: dualism. First there is an object contriving infinitely varied "caricature objects". Then there must be another object that is (infinitely) not any of those. This expression of our relationship to The Great Unknowable Reality is much saner. It doesn't overmine. It doesn't undermine. It doesn't leave me oscillating between affirmation and denial. Also, most importantly, I'm given a clue to further knowledge. I am that contriving object. This is just a caricature of reality. My participation in it's consideration is entirely arbitrary. I'm haunted by the concern that knowledge exists which cannot be captured by this freedom. But for now these objects certainly get us further than nothing. ;)

If world is a set - everything is unique by definition.

You need to define some relation on that set to get classes of abstraction. And that's exactly what abstraction means :)

Even math has an infinity of infinities - Cantor findings etc.

It's a good thing then that most people don't use it in its technical sense

An easy fix for this not problem is to use distinct instead of unique

Curiously (but perhaps only to me) all things are comparable - able to be compared.

If you mean similar, may I commend the word similar? ;-)