yes... the zone will find the minimum elevation... let's say you have a square zone and a flat DEM... the square zone will return the elevation of the flat surface, which is the minimum. This value will be assigned to the zone (which is a raster by the way), that zone will undoubtedly consist of many cells, when the zone is converted to points, you will have many points with the same value.
I would draw a picture, but looking picture of a small sample of your data would even be better given that the chances of every cell within the zone having unique elevations are pretty slim unless you are recording elevation to 4,5,6+ decimal points (which would be just plain wrong)