Anyone who has needed just a handful of small fasteners will recognise this situation. The maker had a small job calling for 16 rivets and made them at the bench from copper and rod, with nothing more exotic than a vice, a hammer and a block of steel involved.
The heart of it is simple. A short piece of copper wire sits upright in a small hole in the steel block, and repeated strikes from a steel ball-peen hammer flare the exposed end until it mushrooms into a flat head, leaving a finished rivet with a broad head on a round wire shank. Alongside that, 1/16 inch tig rod was clamped in a vice, and pieces of filler rod cut to roughly 5/8 inch had a head peened onto one end. That first head is then backed up, the surplus shank is clipped away, and the remaining end is peened shut. Working the second end last, after backing up the first, seems like tidy sequencing to me.
The steel block with its small hole is a nicely low-tech way of keeping the pin upright while it is being struck, and the whole setup stays down to a vice, a hammer and some rod, which I think is part of the appeal here.
One thing I am left wondering about: whether the finished rivets did the job they were cut for. The account ends at the final peen.