A new pyramid-like shape always lands the same side up

That's not a Platonic solid. Come on, like.

Yeah. I tried to google what's Platonic solid and each face of a platonic solid has to be identical.

It's a meaningless distinction. A solid is defined by a 3D shape enclosed by a surface. It doesn't require uniform density. Just imagine that the sides of this surface are infinitesimally thin so as to be invisible and porous to air, and you've filled the definition. Don't like this answer, then just imagine the same thing but with an actual thin shell like mylar. It makes no difference.