@nick-van-den-broeck what you are doing is replacing the geometry of the individual lines with the (linked) geometry of our model.
Indeed; when you select the default Cube and then SHIFT-select a Monkey and then press CTRL+L to link the Object Data, the Cube will look like Suzanne.
Yes, the only problem with the Origins is, that when you put them at the individual Edges, they all have ‘Global orientation’ (Z-axis up).
I love the V shortcut 😉