quantum mechanics - Tensor product postulate

Non relativistic quantum mechanics assumes that a composite system should be described with the tensor product of the component systems. This is the tensor product postulate of quantum mechanics.

I think that the postulate originated in wave mechanics due to the following isomorphism: $\ L^2 \ (R\times R)=\ L^2 (R)\otimes \ L^2 (R)$. The lhs of the former equation is quite intuitive, the geometry of Hilbert spaces do the rest.

Now, disregarding position representation and taking for example two simple quantum systems (e.g a qubit and a qutrit) why, in principle, should we describe the composite system according to the tensor product postulate?