The entries of $\begin{bmatrix}a&b\\c&d\end{bmatrix}\begin{bmatrix}a'&b'\\c'&d'\end{bmatrix}=\begin{bmatrix}aa'+bc'&ab'+bd'\\ca'+dc'&cb'+dd'\end{bmatrix}$ are curiously given by the entries of the composition of rational functions $\frac{ar+b}{cr+d}$ and $\frac{a'r+b'}{c'r+d'}$ which yields $\frac{(a a' + b c')r + a b' + b d'}{(c a' + d c')r + c b' + d d'}$.
Does this have a generalization to $n\times n$ matrix multiplication?
Yes, it does, using the idea of homogeneous coordinates, or equivalently, projective space. Two vectors are called projectively equivalent if each is a non-zero scalar multiple of the other. Multiplication of a vector by a square matrix is a function which preserves projective equivalence. Multiplication of two square matrices corresponds to the composition of the two corresponding functions.
For example, the function that corresponds to multiplication by the matrix: $$A=\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}$$ is $\quad f_A([x:y:z]):=[ax+by+cz:dx+ey+fz:gx+hy+iz]\quad$ where $[x:y:z]$ is a notation for the equivalence class of the vector with coordinates $(x,y,z).$
