An awfully simplistic answer: we work on two-dimensional paper, so two-dimensional matrices are very convenient to write down and compute with, while higher-dimensional hypermatrices are not.
So while we could represent multilinear forms, tensors, etc. as hypermatrices, we often don’t, because doing so is not nearly as fruitful as representing linear maps, bilinear forms etc. as matrices. Instead, we usually use other notations when working with higher tensors by hand.
In computer algebra, the dimension of the paper is not significant, while some kinds of abstraction are harder, so in this context, higher tensors are much more often represented as hypermatrices.