Is it possible to make a selection in Photoshop CS6 and move that selected area on more than one layer at the same time?
Unfortunately I don't think this is possible.
The only way you could accomplish this is by merging the layers together and then making your selection and moving it.
You can move multiple layers entirely together, but not sections of them, much like you can't paint, erase, or do any other action on multiple layers at the same time.
What you can do though, as a precise workaround is make your selection, then go to
Select > Save Selection and save your current selection.
Then finish your selection (moving your section over to where you want it) and then save this new selection.
Then for any further selections on other layers simply go to
Select > Load Selection and load the first selection you made. Now
Cut your selection and load up your second selection and
Paste. Then simply re-merge those two layers. Continue repeating that on every layer you need to.
This process will ensure that every selection starts and stop at the right spot every time.
To better this process you could turn all of this into an
Action and then simply run that action on each layer.