Register images containing objects with varying distance to the cameras
Yes, if there is a lot of variability in the distance to the objects in the scene, then global registration will work poorly.

Let me back and give some background on how global image registration works.  It tries to align the two images.  In its simplest form, the only transformation that is allowed is translation: it tries to find a way to shift all the points in the left image in a constant direction, to make it match the right image as much as possible.  In other words, it looks for a translation $T$ such that $T(I_1)$ matches $I_2$ as much as possible, where $I_1,I_2$ are the two images.

With this background, you can see why global image registration won't work well for scenes with objects at multiple distances, when you have two pictures taken from a different point.  Imagine taking one picture, then moving left one mile, then taking another picture.  Objects that are far away won't shift their location in the image very much, but images that are close will shift to the right a lot in the second image.  Global image registration tries to find a constant translation that makes everything match up in the two images, but there is no single translation that will work well.  To make the distant objects match up, you'd need a small translation; to make the nearby objects match up, you'd need a large translation; and there is no way to reconcile these differences.

This is the major problem you're seeing.  There are other problems as well, due to the fact that the image is only a 2D representation of a 3D scene.  For instance, if you take two photographs from two different positions, then not only will the location of the objects in the images shift, but also you'll be viewing them from two different angles.  Global image registration can't un-do that, because it can't see around corners.  Look at the two images of the statue.  Imagine that the statue had a little bit of bird dropping on the left side of the person's nose (our left; the person's right).  Then the bird dropping might be visible from the second photograph, which is taken straight-on, but might not be visible from the first photograph, which is taken from an angle.  Alignment can't fix this, as it can't magically make up for things that aren't visible in the photograph.



This answer talks about global image registration based purely on translation.  In general global image registration typically allows a broader set of transformations: a typical example might be something that allows translation, scaling (zoom in or out), and rotation; or translation, scaling, and skew; or an arbitrary linear/affine transformation; or an arbitrary homography / projective transformation.  Unfortunately, none of these really help with the problems outlined above.  In general, no rigid transformation will eliminate the problems I've sketched in this answer, so this answer is pretty general.  In principle, non-rigid transformations might be more effective at handling variation in depth; I don't have any experience with that form of image registration.