$\begingroup$

The tensor product makes perfect sense! It is the inner product that does not! These vectors "live" in different Hilbert spaces, you can't make an inner product out of those, it does not make sense.

What the equation means is simply the statement of 2 different particles with 2 different Hilbert spaces. Particle A has its state vector in Hilbert space A and same for B, so $\vec{S}_A$ acts on the states of A and same for $\vec{S}_B$ on B.

Since $\vec{S}_A=(S_x,S_y,S_z)_A=\frac{\hbar}{2}(\sigma_x,\sigma_y,\sigma_z)_A$ and $\vec{S}_B=(S_x,S_y,S_z)_B=\frac{\hbar}{2}(\sigma_x,\sigma_y,\sigma_z)_B$, the tensor notation is the formal way to state that the first operator (S_A) will act on the A state vector and S_B on the B state vector. One simple example would be one component of the tensor product acting on an entangled state of A and B in the z-direction basis of these. So when operating on a state like:

$\left|\frac{1}{2} \frac{1}{2}\right>_A\otimes\left|\frac{1}{2}-\frac{1}{2}\right>_B+\left|\frac{1}{2} -\frac{1}{2}\right>_A\otimes\left|\frac{1}{2}\frac{1}{2}\right>_B$

each operator only acts on the corresponding state.