1) Fields have algebraic closures unique up to a non-unique isomorphism.
2) Nice spaces (without base point) have universal covering spaces unique up to a non-unique isomorphism.
3) Modules have injective hulls unique up to a non-unique isomorphism.
Such situations can lead to interesting groups - the absolute Galois group, the fundamental group, and the "Galois" groups of modules introduced by Sylvia Wiegland Wiegand in Can. J. Math., Vol. XXIV, No. 4, 1972, pp. 573-579.
I'd appreciate any insight into the abstract features of situations which give rise to this type of phenomenon. And I'd appreciate as many examples from as many parts of mathematics as possible.