One of my professors told me that a state in a classical system is an element of a set, while in quantum it's an element of a vector space. From that, we combine systems in classical physics using a Cartesian product, while in quantum using a tensor product.
I'm confused, don't we describe everything in classical mechanics having a position and a velocity, which is a vector, is it?
Often the configuration space (of generalized positions) in classical mechanics is taken to be a differential manifold $M$ (which is not necessarily an affine space). The generalized velocities are then tangent vectors in the tangent bundle $TM$. In other words: Generalized velocities are vectors, but generalized positions need not be.
Well, the difference might not be that sharp. the Koopman-von Neumann formalism describe classical (hamiltonian) mechanics as vectors in a Hilbert Space, and quantum mechanics can be described as a dynamics in a symplectic/phase space.
