Even though there's no shading, a bad resolution and zero context anchoring it in a familiar setting you immediately see it as a 3D object when effectively that's just 3 2D quadrilaterals changing colors and shape. We really need very little to be able to extrapolate depth information from a very limited 2D-only display.
If you did that with 4D cues in 3D space (say in VR or something) I think it wouldn't work. It'd just look like a 3D object changing shape, not a projection of a higher dimension (the same way an hypothetical 2D being would see 3 quadrilaterals changing shape in the above video, not a solid 3D object that would be an abstract concept to them, not a familiar reality).
Could we teach our brain to "think in 4D" and interpret these cues differently? To a certain extent probably, maybe if you spent hours and hours and hours in a VR simulation with 4D objects you'd start getting a feel for it. I doubt you'd ever get as good as with 3D objects since we've been exposed to these literally every waking moment since birth, but maybe I underestimate the plasticity of our brain.
That actually leads me to an other question: is 3D somewhat hardcoded in our brains or is it purely learned? If we were mad scientists and took a baby brain into a 4D "plato's cave" style simulation, could it grow into being able to perceive 4D as intuitively and effectively as we do with 3D space? Also unrelated question: does anybody have a baby I could borrow?