MAD=0 means that the median of the data's absolute deviation from the median is zero. This does not imply that all the data points are the same or that the variance is zero. What it does imply is that zero is the "middle" absolute deviation, and since there can't be an absolute deviation any lower than zero, at least half of the data points have zero deviation from the median - at least half the data points are the median. Since at least half of your data points have zero deviation from the median, the MAD must also be zero.

MAD is just one measure of variability. A MAD of zero does not imply that all other measures of variability, like the variance or mean absolute deviation, will also be zero. Most measures of variability will take a value of zero when applied to invariant constant data, but the reverse isn't necessarily true - some measures, like MAD, may take a value of zero without implying invariance.