Do these higher-dimensional analogues of Möbius transformations have a name?

Solution 1:

These seem to be projective transformations / homographies / collineations. See particularly the formulas given when projective spaces are defined by adding points at infinity to affine spaces.

This is no surprise since there is a long history of projective geometry in optics, going back to the study of perspective. I think you are probably already aware of this, but these maps provide a good description of image transformations by lenses only in the paraxial approximation.

Here's a chapter by Douglas S. Goodman from the Optical Society of America's Handbook of Optics which contains a discussion of these transformations in Section 1.15 (page 59 of the PDF, page 1.60 in the internal numbering of the book). It seems the preferred terminology in optics is "collineation"; note however that Wikipedia distinguishes collineations from homographies, though they agree for real projective spaces.