Caveat: I'm only starting to work through this, so I can't speak will full confidence about it.
It's not about quantum mechanics per se, but rather the application of certain mathematical techniques from QM to at least a certain class of stochastic systems - namely reaction networks / Petri nets, so it at least applies to these Markov processes.
They call a probability distribution a stochastic state, and pursue the analogy with quantum states. Both of these types of systems are driven by a Hamiltonian operator which gives the law of motion.
Ideas like annihilator and creation operators are taken from the quantum context and applied in _this_ stochastic context. For this to work the stochastic state for a N-species reaction network is represented by a power series in n-variables, where the coefficients are the probabilities - which they describe as a "trick" taken from quantum mechanics. So then the annihilator is given by a derivative operator on power series.
They show how the Hamiltonian for a stochastic reaction network can be formulaically constructed by transforming its graph and coefficient structure into a formula involving annihilator and creation operators. The usages of the annihilation operators coincide with inputs to a transition, and the usages of the creation operators coincide with outputs to a transition.
There's more that this, as the book goes on, but this gives some sense of it.
I would describe it as exploratory foundational work.