Can't vector division be defined to give a family of vectors? Dot division of scalar $s$ by a vector $\vec{A}$ gives a family of vectors $\vec{B}$ satisfying $\vec{A}\cdot \vec{B}=s$
Cross division of $\vec{A}$ by $\vec{B}$ gives a family of vectors $\vec{C}$ satisfying $\vec{B}\times \vec{C}= \vec{A}$
Why does the answer need to be unique? Trigonometric equations give a family of solutions. Square root gives two solutions. Integration gives a family of solutions.
 A: 
Trigonometric equations give a family of solutions. 

Inverse trig functions give a family of solutions, but in most cases we pick a convenient branch and stick to it.

Square root gives two solutions. 

When taking the square root of a nonnegative real number, we almost always choose the principal root.  We would probably do the same for complex numbers if there were a nicer way to pick just one root.

Integration gives a family of solutions.

Antidifferentiation gives a family of solutions.  But we really only use antidifferentiation as a way to figure out what a specific integral will be or equivalently we provide an initial condition so that we can narrow the solution down to one number.
The point is, yes there are some things in math that give more than one output for each input (one name for such an object is a multimap), but whenever possible we make some consistent choice to turn that multimap into a regular function.
That said, if you wanted you could define dot division and cross division as you suggest.

Aside: In geometric algebra we have a notion of vector division (or technically just inversion because vector multiplication is not commutative in GA) that gives a unique inverse for every nonzero vector.  Because we have $\vec v^2:= \vec v\cdot \vec v$ for all $\vec v$, we have $$\vec v^{-1} = \frac{\vec v}{\|\vec v\|^2}$$ This works because $$\vec v\frac{\vec v}{\|\vec v\|^2} = \frac{\vec v^2}{\|\vec v\|^2} = \frac{\|\vec v\|^2}{\|\vec v\|^2} = 1$$
