Is $y[k] = y[k-1] + x[k]$ an integrator?
The system

$$y[n]=y[n-1]+x[n]\tag{1}$$

is an ideal accumulator, i.e., it computes the cumulative sum of the input samples:

$$y[n]=\sum_{k=-\infty}^nx[k]\tag{2}$$

It is in a way analogous to a continuous-time integrator, but this doesn't mean that you will necessarily obtain an ideal integrator by transforming the discrete-time system to a continuous-time system. There are several methods to do the conversion, and I'm not sure which one you used when calling d2c.

In any case, the properties of the transformed continuous-time system not only reflect the properties of the original discrete-time system, but also the properties of the transformation. So the properties of a discrete-time system should be investigated directly in the discrete-time domain, not by transforming it to a continuous-time system.