[sigmoid10]

>The tensor product itself captures all ways that basic things can 'interact' with each other!

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.

[hackandthink]

"Product states are multipartite quantum states that can be written as a tensor product"

"In the special case of pure states the definition simplifies: a pure state is separable if and only if it is a product state."

https://en.wikipedia.org/wiki/Separable_state

[sigmoid10]

Keyword is can. There's a reason why most university level physics curricula defer quantum density matrices (and by extension tensor algebra) to more advanced classes. There's a lot of mathematical legwork required before you can actually make use of that.