I used to work with silver and gold, and find that setting the stones in a leather is quite fun. More importantly, a metal setting is not absolutely required. I basically set them in a leather bezel. After cutting a hole slightly smaller than the stone, the leather is wet-formed around the stone by pressing with polished wood tools. The stones can be epoxied and then the bezel is contact cemented and stitched. This can be done with almost any leather thickness, but I generally use 5oz vegetable tanned leather (tooling leather) or greater for bezels.
So far I have been only doing it with cabochon stones though... But just so you know, even cut gemstones can technically be set in leather using the same wet-forming to smush the leather around them!
You could also use specific properties of leather such as how it sort of shrinks as it dries
One issue to consider is that leather is pliable, so stones set too gingerly could potentially pop out when the item is flexed - lol!
The images you showed appear to be silver jewelry which is most probably riveted to the leather items, or it could be fastened in a few different ways such as with screws or bent prongs.