When I hear about the cascade form of an IIR filter or implementing a filter with biquads, is there a state space model perspective too?

One example of this that I've found so far: "diagonalized state-space form is essentially equivalent to a partial-fraction expansion" (https://ccrma.stanford.edu/~jos/fp/Diagonalizing_S...)

The state space model (like we get when we take a Control Systems course) is much more general.  I.e. the very same transfer function can be implemented multiple ways, cascaded second-order sections is one way, parallel second-order sections is another way.  One big Direct Form 2 is another way.  Transposed Direct Forms are other ways.

Each one of these filter topologies can be represented in a state space model.

Right, but I'm wondering if those filter topologies have nice structured matrix interpretations.

They do.  I would have to find the right textbook or paper.  Maybe Julius Smith has it spelled out.  Look at https://ccrma.stanford.edu/~jos/fp/Converting_Stat...

BTW, for any of those topologies, the state space matrix will have a lotta zeros in it for a high-order filter.  That's because a state in one of the cascaded second-order sections (SOS) is not directly connected to any of the states in another SOS far away, so the coefficient connecting them is 0.

Do you know how to set up the discrete-time state space matrix given a drawing of a filter?  This is not all that hard.  But I'll see if I can find an online source.