Is $y[k] = y[k-1] + x[k]$ an integrator? It looks exactly like an integrator to me.
Since $$y[k] = y[k-1]+x[k] = y[k-2] + x[k-1] +x[k] = \sum{x}$$ 

Applying the Z-transform gives
\begin{align}
Y(z) &amp;= Y(z)\cdot z^{-1} + X(z)\\
\Rightarrow\frac{Y(z)}{X(z)} &amp;= \frac{1}{1-z^{-1}}
\end{align}

When I convert $\frac{1}{1-z^{-1}}$ into Lapace transform in Matlab using d2c, it does not return $\frac{k}{s}$. Instead, it returns $\frac{1+s}{s}$. It seems there is a proportional term inside, when thinking in PID control point of view.

Could anyone explain on that?
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.