The fins there are 2mm polystyrene sheet, variously sold under names like "plasticard" and "sheet styrene". It's not a serious engineering material, but I'm a wargames fanatic and thus have bulk orders of it in different thicknesses because it is good for scratch building vehicles and weapons for models.
Ultimately, I just fixed the fins on with super-glue/kragle/cyanoacrylate/whatever you want to call it. It's strong enough to keep the projectile together during launch (as most of the compression forces are just passed through the fins to the base of the metal core, with little reliance on the shear strength of the glue), fine in flight, and weak enough to break on impact, letting the penetrator core pass through the target unimpeded.
These particular ones are the baby ones, but they still went through 6mm mild steel and several water jugs like a hot knife through butter:
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The large white disk you can see come in from the top of the screen, bounce near the left corner of the backstop and head into the tree is the sabot disk that fits behind the projectile. You can see several fragments of fin splashing out from the initial impact, and the core bounces off the backstop to the left. (Actually, if you watch really closely, you can see the core zoom out like a hornet, catch the edge of one of the concrete blocks, bounce back against the water jugs and then fly off at a more sedate velocity).
Those are 75mm masonry nails (with the heads cut down and the points sharpened) about 3.5mm in diameter. The whole assembly is 20mm calibre and about 95mm long.
(The bigger version is basically the same with a 6mm hardened tool steel core instead).
I'm guessing it's tungsten
Well, check with a magnet first, as a lot of AP cores are hardened steel (which can still blunt tool blades, after all, they're hardened steel too), but even if it is, it's still got the potential to be a very mean projectile.
... and a little safer too. Tungsten is rather toxic if you prove to be one of the unlucky people who finds a way to get it into your system... and a skin irritant too, but that's less significant.
In any case, if it does prove to be tungsten, treat it with respect. (Although don't worry yourself unduly about previous exposure - if you weren't suffering from nausea, sudden seizures or a coma shortly afterwards, you probably didn't give yourself acute tungsten poisoning).