How to remove the fork off this Peugeot Nouveau

I believe the chrome steel collar, this collet type piece that it grips, and the cut out area of the steerer they're sitting on are how the stem is held tight on this bike. The aluminum collet piece may be there so it's an aluminum and not a steel thing gripping the steerer threads. So, it may not be threaded as such, but is pushing up against threads and probably has indendations of threads formed into it. It's an old aluminum-on-steel interface with lots of contact area, so you may be up against aluminum oxide corrosion if it's stuck and could use anything you'd use on a stuck seatpost. My current favorite to make quick work of it are thermal products like Finish Line Chill Zone.

If it's stuck, you may get somewhere by putting a screwdriver or other similiar tool in the slit and twisting it just enough to flex it open.

A strap wrench could also be your friend here because it won't hurt the part and is the ideal way to get leverage in situations like this.

It appears that the top race of the headset is a common French type with knurling on the outside and a toothed washer on top, so this collet and clamp assembly you're working with is likely not part of what adjusts or holds the headset bearing adjustement.