Re: It's just mathematical equivalency
As soon as you start asking questions about whether there is some sort of ultimate reality underlying mathematics, you're into Platonism. AFAIK, mathematicians are always discovering that "this" way of doing something is equivalent to "that" way of doing something, and the different routes are connected as some more fundamental level.
Platonism and "ultimate reality underlying mathematics" means Plato's ideal forms to me.
Mathematics is the matching of patterns and hence it isn't surprising that the same pieces of mathematics can be reused in different areas where the same patterns turn up. Mathematical equivalence of two theories is somewhat different - in that case two apparently very different patterns turn out to really be the same. This is somewhat rarer.
The alegory of the Cave like the parable of the blind men and the elephant points out that our perceptions are not reality. However they are all we have to try and discern reality.
In a sense the holographic principle seems to be Plato's cave in reverse. The two dimensional shadows are supposedly creating the 3D world we see around us. Normal shadows cannot do this - the 2D images need to somehow encode the 3D images. We can do this with holography but only by starting with a real 3D object and using two laser beams to create a 2D interference pattern.
This interference pattern then being later redisplayed as a 3D image when another laser beam is passed through it at the right angle. Having a physical arrangement where the surface of the Universe just happened to have a pattern on it which through some process equivalent to projecting a laser through it generated our 3D world would be impossibly unlikely. However a mathematical equivalence between a theory describing our 3D world and an equivalent theory on a 2D surface would be possible. Why a theory describing our 3D universe would have to have such an equivalence with a theory on a 2D surface seems to me a puzzling question but there could be some underlying reason why the Universe has to be that way - most likely resulting from some symmetry in the universe just like conservation of energy and momentum arise from underlying symmetries.