I mean typically people refer to planes as hyperplanes once you go past 3D, but I’ve definitely heard them just called “planes” too

Hyperplanes are just a generalization of planes to higher dimensions. Often you hear the term when working with vectors because, like in 3D, you can define an n-dimensional hyperplane by a surface normal vector and a point. All lines perpendicular (orthogonal) to that normal vector which pass through the point form the plane.

It’s a useful concept and since we already have a word for that kind of structure in 3D space we just use the same term for it in other dimensions

Hmmm. I don’t think so but I do think it is incorrectly used here. To me, the 3rd dimensional plane would just be the z axis. If you were talking about the entire 3D shape which they are here, I would just say 3rd dimension.