Strange behaviour when setting Origin to Cursor position for a "duplicate linked" obj


I have an original Object (green in the following screenshot). I have another object (red) which I want with a linked duplicate of the original object (the duplicated Object - created with Alt-D - is orange).
The cursor’s where I want the duplicate to be. However, when I choose “set origin to 3d cursor” the result is not what I expect (I would expect the red and orange object to be at the same location).

What am I not getting here and how do I achieve my goal quickly and easily (in my “real” scene, I have very many “red” objects).

MoveLinkedDuplicate.blend (565 KB)

Best regards

Damn it. Almost had it.

In the “Object Data” I just set the red data to the green mesh data block. If that makes sense to anybody, to me it makes.

Sorry for this whole post. (Still I don’t understand the above mentioned behavior)