A deliberately extreme example: an isomorphism of sets is a bijection, and two sets are isomorphic when they have the same cardinality. There is generally no preferred bijection between sets of the same cardinality. For example, there is no canonical choice of bijection between commonly used sets like $\mathbb N, \mathbb Z, \mathbb Q, \mathbb Z^2, \bar{\mathbb{Q}}$.

This ties in to Mark Wildon's example, too: if any two sets of the same cardinality had a canonical bijection between them, each finite set $X$ would have a canonical bijection to a set of the form $\{0, \ldots, |X| - 1\}$. This would give a preferred total ordering on $X$ (the one corresponding to $\leq$), which in turn would give a canonical bijection between $\operatorname{Ord}(X)$ and $\operatorname{Perm}(X)$.