Standard deviation is varying with change in amplitude of DC signal
I believe your intuition is correct.

The filter you're using is a low pass filter with a DC gain of 1. Without seeing the output of the filtering operation, I suspect that you're seeing the effect of initial conditions in the filter: getting from 0 to 1 in your first instance takes fewer samples than getting from 0 to 10 in your second.

To test this, try taking the middle $L/2$ of your filtered signal and see what std(y(L/4:3*L/4) is then.