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 at least half of your data points take the median value, which have zero deviation from 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.