Hi, thanks for playing!
You were right about the 2D object (the octopus and the floor) having hidden 3D colliders. A trick I found while doing this is to make the colliders relatively thick. If these colliders are too thin, there are chances that small movements will cause the player to fall through the floor.
I didn't lock the axes, though, since the axes would rotate when the player rotates with the cube (or in other words, I'm using the player's local axes instead of the world axes; I don't think I locked any of the axes).
I've one script that helps lock the distance between the player and the surface of the cube, and the player also has a custom gravity script that applies gravity to the player's local downward direction (this script is disabled during rotation to prevent the player from falling). When standing on a cube, the player will temporarily become a child of that cube so it can rotate along.