Matrices are excellent for manipulating frames of reference. Those whose members are not restricted a common set of numbers/entities (or are themselves matrices) are extremely useful to me.
Their use in the manipulation of concepts is essential in building knowledge representations.
Are there others who are manipulating matrices whose elements are themselves matrices or other unique entities? If so, how are they doing it?
Assuming to maintain the usual matrix operations, i.e. to obtain a ring, we can formulate a minimum requirement on the structures of the sets where the entries are coming from.
Even in the most common case when all entries are from the same ring $R$, we can allow $R$ itself to be a matrix ring, moreover we can iterate it as well..
But we can consider different sets $A_{ij}$ (with at least Abelian group structure for the addition) holding the possible $i, j$-entries.
Note that the diagonal sets $A_{ii} $ must be themselves rings, but we can relax it somehow for the other entries.
For instance, the generalized matrix rings of the form $\pmatrix{R&M\\ \{0\} &S} $ are exactly those where $R$ and $S$ are rings and $M$ is an $R$-$S$-bimodule.
The $2\times2$ generalized matrix rings $\pmatrix{R&M\\N&S} $ correspond to Morita contexts between rings $R$ and $S$, connecting the bimodules ${}_RM_S$ and ${}_SN_R$.
That's all my knowledge on the topic.