Perhaps another reason not to distinguish between isomorphic structures and isomorphism classes ... Many years ago, Grigorchuk defined a very beautiful space $\mathcal{G}$ of finitely generated groups with the following features:
(i) $\mathcal{G}$ is a Polish space; i.e. a separable completely metrizable space.
(ii) The isomorphism relation on $\mathcal{G}$ is induced by a natural action of a suitable countable group $G$.
However, Champetier has shown that the "quotient space" $\mathcal{G}/G$ of isomorphism types is horribly complex and cannot be given the structure of a decent topological space in any natural manner.
