You can multiply matrices that way, but it turns out to not be as useful in as many cases. Matrix multiplication the way you're taught turns out to be the most useful multiplication-like operation you can do on matrices, so it's the one that just gets the name "product". The other ways you can multiply matrices are not as commonly used, so they get other names, like elementwise (or Hadamard) product, and Kronecker product.
This is a result of what matrices are. There are different interpretations that apply, but here is the one I most often think of:
A matrix is a description of a linear map. It describes this linear map by listing which column vectors the basis vectors are mapped to (the first basis vector is mapped to the first column of the matrix, the second basis vector is mapped to the second column, and so on). And the one multiplication-like operation you can do with linear maps is to compose them, i.e. doing one after the other.
So, if you have two matrices $A, B$, each representing a linear map, then the matrix that represents the composition of the two would deserve the name "the product of $A$ and $B$". And if you work through the geometry and the algebra, the standard matrix product is what falls out.