State space physical meaning
As you pointed out, there are many state-space realizations of one particular transfer function. The reason is that a transfer function only represents the input-output behavior of a system (observable and controllable dynamics) and not the internal states.
That being said, you can directly write state-space realizations from a transfer function with the so-called vertical and horizontal companion forms (equivalently, the observable and controllable canonical forms). The problem is that the physical meaning of the states is not usually guaranteed.
So unless you have a model of your system in terms of differential equations (first principle modeling like Newton's laws or Lagrangian mechanics for mechanical systems), the physical meaning of the states will usually not be accessible. The same holds in the discrete-time domain.