Is it possible for two different systems to have the same step response, and vice-versa?
Is it possible for two different systems to have the same step response, and vice-versa?


Of course. 

Consider a basic RC low-pass filter driven by a low-impedance source and with a high-impedance load. You could have either R = 100 ohms and C = 10 uF, or R = 1 kohms and C = 1 uF and get the same step response.

Or you could construct an LR dual circuit driven by a current source and get the same step response. 

In fact, any first order linear system with the same characteristic frequency will have the same step response.


  if a process is an excellent fit with a model, does that necessarily mean they are nearly equivalent (and therefore controllers tuned to the equivalent model will always work on the real process to some extent)


In my experience there are cases where a first order model appears to fit the step response very well, but the system is actually second-order, and this can affect the ability to achieve good control (when trying to use a controller matched to the first-order model).


  or are some aspects of the system not captured by the step response?


But this doesn't mean the step response did not capture the full behavior. Only that some aspects of the behavior may have only a small effect on the step response, and so be difficult to fit.

On the other hand, of course the mapping from step response to frequency response is only 1-to-1 for linear systems. If your system is nonlinear, the step response will not capture the full behavior.