There is one thing being overlooked here.
A "reflection" not only generally preserves the intensity of the light being reflected, but in the ordinary case it also preserves the phase (other than a pi shift) of that light.
Now, why is this important?
Note that in an ordinary picture that you see in a photograph or a book, that picture is "replaying" only the intensity of the content. When you shine light to it, that is what it is reflecting back into your two eyes. A hologram, on the other hand, replay not only the intensity, but also the phase, or more accurately, the phase difference between the various objects being replayed. So when you shine light onto a hologram, it does not only reproduce the intensity, but also the phase difference between the various objects. When we view such light source, our two eyes and our optical nerve process these into various depth and perspective.
The same thing occurs when visible light is being reflected off an ordinary plane mirror. The phase difference between the different objects are preserved upon reflection. Thus, your eyes can't tell the different (other than the left-right switching) that you are looking at the image from a mirror rather than directly at it.
Zz.