From images and description it looks like you have some solid bodies and some surface bodies.
Duplicate tool works only on visible bodies. Make sure you have only one solid body visible. Then try Duplicate. Does it show any duplicates?
Now make only 2 surface bodies visible which are coinciding. Now use duplicate. Does it show duplicate? If yes, delete one duplicate surface body.
You may also want to check Share Topology option.
Also see Split Body option.
Please go through help manual for more details
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