From a mathematical standpoint, this is now a projective vector space.
Some references:
https://en.wikipedia.org/wiki/Transformation_matrix
https://www.quora.com/Why-does-OpenGL-use-4D-matrices-for-ev...
Meaning we have to be very careful about treating our projective transformations as linear maps, as they are, in fact, nothing like that.
Of course people doing graphics often pick a section of the projective space ("last component is 1"), then map that onto R^3 and off they go, sort of ignoring the "planes at infinity" and praying for the best. This is messy but I do not see a better way.
In your second link, I like the trick of storing positions as [x, y, z, 1] and directions as [vx, vy, vz, 0].
Regarding, "Hey, Markus! How come this matrix is 4x4?", I'm not sure if Markus was referring to an affine transformation, part of a perspective projection, a matrix representation of a quaternion, or something else. The mere fact a matrix is 4x4 (or some other size) unfortunately doesn't say much about what it's meant to do. Which probably contributes to Markus getting so many questions.