Why can't we create new vector multiplications like:
$$\vec{A}\odot\vec{B}=|A||B|\sin \theta$$
and
$$\vec{A}\otimes\vec{B}=AB\cos \theta \, \hat{p}$$
where $\hat{p}$ is a unit vector perpendicular to vectors $\vec{A}$ and $\vec{B}$.
Why don't we get any physical quantities from such vector multiplication?
You can - you can always define new stuff in math (if they're defined properly), but you need to ask yourself is it interesting. Can you make a whole new inner product theory with your 'Scalar Product' (with other implications and theorems)? Maybe. Probably, some people tried to explore that area and didn't find something worth publishing or sharing with the world, or they did and it didn't get a lot of attention.