Can error in digital signal be zero after filtering?
The discrete signal is a sum of two sines. The noise is an additive uniform noise, always positive (rand lies in open interval $(0,1)$). The latter  is the crucial point with the median. Suppose you have a frame of three signals samples: $x_0$, $x_1$, $x_2$, and three noise realizations:  $n_0$, $n_1$, $n_2$ in $[0,1]$. So you are checking if:

$$\textrm{med}\{x_0+n_0,x_1+n_1,x_2+n_2\}-x_1$$

can be zero for each window frame. You have noticed from the plots that the noisy signal is always above the clean one. So the  median of $\{x_0+n_0,x_1+n_1,x_2+n_2\}$  is always higher than the  median of $\{x_0,x_1,x_2\}$. 

But could it be the same as the initial data (putting signal borders away)? Yes. Take  $\{x_0,x_1,x_2\}=\{0,0.5,0.2\}$ and add the random realization $\{0.5,0.1,0.3\}$. Then the median of the noisy signal is $0.5$, the exact value of the center sample $x_1$.

So for some well-chosen signals and noise, their are counter-examples, a median filter can in cases provides a zero error. However, and I admit I cannot give a proof at this time, it is probably very unlikely that there is a zero error for  long enough generic signals.

Note to DSP folks: I feel like I am missing a very evident proof, or such a question is not adapted at this level of signal processing.