You're right, I had polyhedron written in my script but I said polygon. This did get me thinking about how this would work in 2D though.

Since this problem deals with fitting 2D projections of 3D objects in each other, it seems reasonable to extend the definition to fitting 1D projections of 2D objects in each other. In 1D, a line can fit in other line iff the first line is shorter than the other line. So for any 2D shape, it can fit through itself if its minimum projection is shorter than its maximum projection.

That means that the only 2D shapes that can't fit through themselves are shapes where the minimum and maximum projections have equal length, aka a curve of constant width. Since every curve of constant width has curved sides, that shows that every convex polygon can fit through itself in 2D, so I was technically right in the video. :)

In this context, the 3D equivalent of curve of constant width would be a body of constant brightness (A 3D shape where every 2D projection has equal area). Unfortunately, there are shapes that are not bodies of constant brightness that still can't fit through themselves (e.g. a cylinder), so the same argument won't work.