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.