According to the definition of h, "if h of his/her N papers have at least h citations each, and the other N − h papers have no more than h citations each." Looks like the h value is unique. The "Note" part looks unnecessary.
Proof: say there exist two h-indices, h1 < h2 (1), which means that, there are h2 numbers have value at least h2 (2) and N-h2 values have value at most h2 (3); there are h1 numbers have value at least h1 (4) and N-h1 values have value at most h1(5).
We have (1)(2)(3), (1) and (2) implies (4), but we can see (5) is not possible. Because we have N-h1 > N-h2, there exists at least one paper not in the set of N-h2 but in the set of N-h1, this paper should have value <= h1 (according to 5) and have value >= h2 (according to 2), this contradicts to (1).
The uniqueness follows.