In quantum physics, "interacting" usually has a different meaning. So one should use these terms more carefully.

>And you need the tensor product already for pure states in QM.

No. Pure states are just vectors (or more precise: rays) in Hilbert space. The usual inner product is sufficient to work with them. An outer (=tensor) product of these states will just give you a density matrix with tr(ρ^2)=1.