Yes, the quantumness of a differential equation is not a property of the differential equation, but a statement about one possible taxonomy of differential equation. Then, whether quantum-type diff eq's have unique properties pertaining to chaos conditioned on our knowledge of them being labelled 'quantum' is an interesting mathematical question.

> Note that there is no Lagrangian for the NS equations, by the way.

I don't know much about fluid dynamics, but I was under the impression that Bennett derives the Lagrangian form in the book Lagrangian Fluid Dynamics