One example that springs to mind is when you are secretly working with the objects of a higher category, and so the choice is not unique up to a unique isomorphism, but the choice of isomorphism is also subject to higher coherence data. In a 2-category this would mean the isomorphisms are unique up to a unique invertible 2-arrow and so on. In $\omega$-categories, you may have such coherence all the way to infinity, and so end up with no uniqueness after all. One place where this emerges is when your $\omega$-category has all duals - is then an $\omega$-groupoid, because all the duals make everything weakly invertible!