In scientific programming, this can be quite common. We just start calling these tensors of large dimensional arrays. Scalars are 0-dimensional tensors, vectors are one-dimensional tensors, matrices are two-dimensional tensors, and after that we call them d-dimensional tensors (d = 3,4,5,6). Point products are called abbreviations for indexes.
Where are they used? I use them in some of my physical simulations. For example, one method of modeling electrons on a lattice (a regular array of sites) uses a tensor with a different set of indices for each connection to a neighboring site. In a two-dimensional square lattice (count the places in the center of each space on a chessboard), this means that each tensor has four indexes: one for each neighboring site, so it is a four-dimensional tensor.