Yes, the distinction both exists and it doesn't. When defining things, you might start off by saying "the natural numbers are von Neumann ordinals". Then you construct the integers as certain infinite sets of pairs of natural numbers, and you say "actually when I say natural numbers I mean integers that contain a pair where the 2nd number is 0". Then you define rationals as certain infinite sets of pairs of integers, and say "actually when I say integers I mean rationals that contain a pair where the 2nd number is 1", and so on. So for a brief moment during the construction of the next step, there is a distinction. Then you immediately retcon your definition and get rid of it. No one ever uses the intermediate definitions again.

There's similar logical snags when trying to define real numbers because technically you'll need distances which have to be rational because you don't have real numbers yet, but really you'd like distances to be real. It's not actually an issue though, and as far as everyone is concerned, distances are real.

Or you define things only up to unique isomorphism by their properties and wash your hands of the whole ordeal. The construction is merely to show that some object with those properties exists.

The teacher is wrong because if they are being pedantic about it to a child, they're a bad teacher. And they're missing the point.