Is there an existing name for "piecewise vector multiplication"

That is the Hadamard product---which usually, though, is only used with matrices of a more matrixy shape.

If $k$ is a field, the vector space $k^n$ endowed with the componentwise multiplication is called a diagonal algebra ( and so are isomorphic algebras). The terminology is due to Bourbaki and is justified by the following result.

If $M$ is a square matrix over $k$, it is diagonalizable over $k$ if and only if the algebra $k[M]$ is diagonal.The proof results from the diagonalization criterion (the minimal polynomial of $M$ should be split over $k$ and have distinct roots) , the isomorphism of $k$-algebras $ \frac {k[X]} {polmin_M (X)}\to k[M]$ and the Chinese remainder theorem.

These algebras are important to algebraic geometers because they are a model for étale algebras over a field. Indeed, a $k$ -algebra $A$ is étale if and only it becomes diagonal after some extension of the base field . More explicitly $A$ is étale if and only if for some field extension $K/k$ we have an isomorphism $A\otimes_k K \simeq K^n$ of K-algebras.

It's pointwise product. See Wikipedia articles here and here