Agreed. Axioms are created, everything else is a property of those axioms and 'discovered.' Those basic axioms tend to be the most useful in our universe, but they were still 'created.' We often define or create other axioms, for example, taxicab geometries are imminently useful for modelling paths in cities. We created the rules for the taxicab geometry, as with Euclidian geometry, but we didn't create the properties that emerge from those rules. I think maybe the confusion comes from a misunderstood analogy: people think 'if you build a castle of LEGOs, you created that castle; aren't maths likewise created?' The misunderstanding is that mathematical theorems and properties aren't like the castle, they're like the potential to build a castle from those given LEGOs. The child created the castle, but she only discovered the possibility of building a castle, which always existed as a property of those LEGOs.
I think mathematics in terms of physics, because that's where I did my heavy learning. So when I think 'operator' I think in terms of QM operators, which are defined actions, instead of assumed properties. I see the two as separate, although I suppose they probably seem less so when viewed through a purely mathematical lens.