My understanding of Milne's comment is as follows (note: my history here is second-hand, so it may contain mistakes): when class field theory was first proved, it was not by actually producing an isomorphism between class group and Galois group, but rather by checking that they had the same number of elements of any given order. Apparently, for many years it never occurred to anyone to find a particular isomorphism, but we now know that one exists. In retrospect, it seems outrageous (and thus worthy of comment) that no one seemed to be bothered by the lack of an actual constructed isomorphism at the time.

In fact, there are examples out there is mathematics of groups which are provably isomorphic, but don't have any preferred isomorphism (a finite abelian group and its Pontryagin dual, for example).

My understanding of Milne's comment is as follows (note: my history here is second-hand, so it may contain mistakes): when class field theory was first proved, it was not by actually producing an isomorphism between class group and Galois group, but rather by checking that they had the same number of elements of any given order. Apparently, for many years it never occurred to anyone to find a particular isomorphism, but in fact there is one which most observers agree is "canonical". In retrospect, it seems outrageous (and thus worthy of comment) that no one seemed to be bothered by the lack of an actual constructed isomorphism at the time.

In fact, there are examples out there is mathematics of groups which are provably isomorphic, but don't have any preferred isomorphism (a finite abelian group and its Pontryagin dual, for example).