The 'hologram' in holographic physics doesn't refer to the standard 3d optical illusion things. It's a more general term for 3-dimensional information being encoded on a 2-dimensional surface. Just like the hologram you're thinking of constructs a 3d image out of a bunch of 2d dots, so the entire physics of our 3d universe are constructed out of information on an imaginary 2d surface.
The optical holograms we see often are an example of holograms in general, but they're far from the only ones.
Indeed, it's amazing tech. It's even better than you think: size of the image is dependent on projection, not size of the medium. If you cut it it will show the same size as before, but with lower resolution. That is one of the things that blows my mind every time: the less data points there are, the less precise it is, but it's all still complete.
The optical holograms we see often are an example of holograms in general, but they're far from the only ones.