The equivalence classes referred to by Wikipedia don't correspond to order types; different orderings belong to the same equivalence class so long as the underlying sets have the same cardinality.
To elaborate: you can define an equivalence relation on unordered sets by saying that A~B if A and B can be ordered equivalently. This is, of course, an equivalence relation on sets, so of course it partitions the class of unordered sets. It just doesn't do it in the nice way you assumed it did, where different order types end up in different classes.
