I have 2 identical islands stacked together. Blender doesn't merge the vertices that are in the exact same position when I use Remove Doubles or Weld. It only works when the vertices are NOT in the same exact position.
Here's a GIF:
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I've tried this, and even after doing the welding, most of the time, it is temporary (maybe I am doing something wrong - Oh Well).
Regardless of this, the easiest solution to this is to turn on proportional editing O, and scroll the influence area way down. This way when I move one vert, then everything directly under/over my active vertex comes along for the ride.