The claim usually made is that Metatron's Cube contains all five Platonic solids. The precise version of that claim is narrower, and more interesting, than the loose one.
What the figure contains are two-dimensional projections: subsets of the seventy-eight lines which, read as a flat drawing, resemble the standard isometric view of a solid. Nothing three-dimensional is present in the figure itself. A tetrahedron, an octahedron and a cube can all be traced convincingly, because their isometric outlines are triangular or hexagonal and the underlying point lattice is triangular. Those three are a genuine fit.
The icosahedron and dodecahedron are not. Their vertices in a true isometric projection do not land on the thirteen centres, and this has been demonstrated rather than merely asserted. To make the tracings correct, the icosahedron has to be scaled and rotated relative to the isometric cube, and the dodecahedron scaled and rotated differently again; in the standard drawing the dodecahedron sits inside the cube without touching it while the icosahedron pushes out beyond it. The versions printed in most popular books and on most posters are therefore approximations that look right at a glance and fail on measurement.
This is a live dispute inside the sacred-geometry literature rather than an attack from outside it, and the corrections proposed are geometric rather than doctrinal. The defensible statement is that three of the five solids project cleanly onto the thirteen centres and two do not without adjustment. The indefensible one is the flat assertion that the figure contains all five.