You need to use the Vertex->snap-together-by-distance function. This will move 'close' vertices together.
When duplicating, the new object is moved by a grid unit - you can switch this off in File->settings->General->offset-duplicates. Alternatively, use Edit->copy/paste (control C, control V) to make an exact copy in the same location (although it can be difficult to see that there are two objects in the same location, which is why duplicate shifts the new object)
Andy
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