Kittel and Kroemer derive the pressure of a statistical state in the following way:
They assume a volume compression of a system such that the quantum state of the system is maintained at all times; thus, the entropy $(\sigma)$ is constant in the process of compression. Now let the energy of the system with state $s$ be $\epsilon_s$, then
$$\epsilon_s(V-\Delta V)=\epsilon_s(V)-\frac{d\epsilon_s}{dV}\Delta V+...$$
Since the work done on the system is equal to the change in energy of the system, $$U(V-\Delta V)-U(V)=\Delta U= -\frac{d\epsilon_s}{dV}\Delta V$$
In terms of pressure, $p_s$ on the state we can write $\Delta U = p_s\Delta V$, thus getting
$$p_s=-\frac{d\epsilon_s}{dV}$$
Defining average pressure, $p=\langle p_s\rangle $
$$p=-\bigg(\frac{\partial E}{\partial V }\bigg)_{\displaystyle\sigma}$$
where $E$ is the average energy of the system.
Now, it makes sense to me to talk about average energy, since it's just the sum of all possible energies divided by the number of them. The same would have been the case with pressure, but the fact that we are defining it in terms of such precise and singular conditions (isentropic compression of one particular state at a time) makes me have trouble understanding its physical meaning specifically, on how the actual pressure measured is related to $p$, given the specific condition to measure $p_s$.
