This is just low brow philosophical sounding rubbish of the same variety as "what is 'is'" nobody knows. Sounds profound though.
Matrix is just one way to organize data. When linear operators are organized this way composition of linear operators map to matrix multiplication.
But that is just one of the ways that multiplication may be defined on matrices, Hadamard products, Tensor product, Khatri-Rao product are some of the other examples. They all correspond to different mathematical structures one wants to explore or use. If linear algebraic structures is what ones to explore or use then one gets matrix multiplication.
Matrix is just one way to organize data. When linear operators are organized this way composition of linear operators map to matrix multiplication.
But that is just one of the ways that multiplication may be defined on matrices, Hadamard products, Tensor product, Khatri-Rao product are some of the other examples. They all correspond to different mathematical structures one wants to explore or use. If linear algebraic structures is what ones to explore or use then one gets matrix multiplication.