In general, there isn't a unique tensor (monoidal product) on a category. For example, any category with binary products and coproducts has the structure of a monoidal category coming from each.
There are two (fairly) simple ways to force the tensor to be unique, however.
First, a method that doesn't really introduce anything you haven't seen, but maybe just punts the question. You may be familiar with the idea that the set of linear transformations from $A$ to $B$ can itself be given the structure of a vector space (call it $[A, B]$). Given such an internal hom (which is functorial in its arguments), one can define the tensor of two objects to be an object $A \otimes B$ such that $\hom(A \otimes B, C) \cong \hom(A, [B, C])$. Phrasing this a universal property, we should have a morphism $\varphi: A \to [B, A \otimes B]$ such that for any morphism $f: A \to [B, C]$, there exists a unique $g: A \otimes B \to C$ such that $[B, g] \circ \varphi = f$.
It turns out that for technical reasons, for this to have good properties (like associativity), this needs to be upgraded to an isomorphism $[A, [B, C]] \cong [A \otimes B, C]$, but for the case of vector spaces, that's not much harder to prove.
So what does this have to do with multilinear maps? It turns out a multilinear map $A \times B \to C$ is the same as a linear map $A \to [B, C]$, so saying that these in turn correspond to linear maps $A \otimes B \to C$ is simply expressing the universal property above.
A more principled way to do this requires that we generalize categories to multicategories. A multicategory is like a category, but now our domains are finite lists of objects. That is, a morphism can go from the list $(A_1, A_2, ..., A_n)$ to an object $B$. For the case of vector spaces, we can define the maps $(A_1, A_2, ..., A_n) \to B$ to be the multilinear maps $A_1 \times A_2 \times ... \times A_n \to B$. (Note that in the special case where $n = 0$, this is simply an element of $B$, or more precisely, a function from a singleton set to $B$ with no linearity requirements).
Then the tensor product on this multicategory, if it exists, is an object $A \otimes B$ together with a (mutli)map $\varphi : (A, B) \to A \otimes B$ such that for any map $f : (A, B) \to C$, there is a unique map $g : A \otimes B \to C$ such that $g \circ \varphi = f$. Put another way, there should be a natural isomorphism $\hom((A, B), C) \cong \hom(A \otimes B, C)$.
The properties of multicategories (see the link above) ensure that this tensor is well-behaved, including associativity. If you introduce an empty tensor (an object $I$ such that $\hom((), C) \cong \hom(I, C)$), this empty tensor behaves as a unit for the tensor product ($A \otimes I \cong I \otimes A \cong A$).