Is there generalization like tensor product for bilinear product in algebra to category theory Many algebra texts introduce the temporary term bilinear product followed by tensor product with the proposition such as
If k is a commutative ring and $A$ and $B$ are $k$-modules, then the $k$-module $A \otimes_k B$ is a k-bilinear product.
Or the theorem such as
To each $k$-bilinear function $h: A \times B \rightarrow C$, there is exactly one $k$-linear transformation $A \otimes B \rightarrow C$ with $t(a \otimes b) = h (a,b)$
Is there a categorial generalization for this phenomena to replace a temporal notion like bilinear product with tensor product ?
 A: Yes and no. As Daniel Schepler points out, the monoidal product in a monoidal category is usually taken as the general notion of a "tensor" product. But this is just an axiomatic specification of what the "tensor" product is, and it is non-canonical: there can be multiple monoidal structures on a given category. The monoidal product in a monoidal category is not described by a universal property. This is unusual and unsatisfactory for a basic categorical construct.
Indeed, it's too unusual and unsatisfactory to be true. The characterization of the tensor product in terms of the set of bilinear functions generalizes. The key is that the set of bilinear functions is just a particular hom-set in the multicategory of multilinear functions. Given a multicategory there is, up to a suitable notion of equivalence, a unique monoidal category corresponding to it. In that monoidal category, the monoidal product is characterized by a universal property involving the multicategory, namely: $\overline{\mathcal{C}}(A\otimes B;-)\cong\mathcal{C}(A,B;-)$.  This suggests that the multicategory should be taken as the more fundamental things in these situations with the monoidal product just a way of embedding into an ordinary category so we can conveniently reuse ordinary categorical notions.
As a bit of a tangent, the category of bimodules over $k$-algebras where $k$ is some commutative ring ($k=\mathbb{Z}$ gives bimodules over rings) actually forms a pseudo double category where horizontal composition corresponds to the tensor product of bimodules.  The previous paragraph suggests that it might be better to consider a virtual double category instead.
