Any two injective resolutions (of an object in an abelian category) are homotopy equivalent, but this homotopy equivalence is not unique. This is of course because the lifting property in the definition of "injective" does not require any uniqueness.

Any two injective resolutions (of an object in an abelian category) are homotopy equivalent, but this homotopy equivalence is not unique. This is of course because the lifting property in the definition of "injective" does not require any uniqueness.

The connected sum of oriented manifolds is unique up to homeomorphism, but this homeomorphism is not unique.

A bit silly, but: In a short exact sequence $0 \to A \to B \to C \to 0$ in a semisimple abelian category $B$ is unique up to isomorphism (namely, $B \cong A \oplus C$), but the isomorphism is not unique.