This is an embedding diagram. It illustrates the curvature of a 2D manifold by embedding it in a manifold of higher dimensionality - in this case 3D.
But this is just an illustration. The extra dimension doesn't exist in general relativity. Instead the curvature of the manifold is intrinsic. Because the extra dimension doesn't exist occupants of the universe cannot move into it, so in the diagram you show the observers are confined to the sheet and cannot move out of it into the third dimension. An observer falling into the black hole follows the trajectory within the sheet I've shown in red:

So it isn't possible to move out of the sheet to the middle of the throat and then stretch out your arms to toch the sheet at either side, because the throat doesn't actally exist.