Perhaps the most obvious is its a form of inductivism, as it supposes that you can build representations of the targets of the measure domain from the measurement domain itself.
This is just the same as drawing lines around patterns of stars, calling them "star signs" and then believing they are actual objects.
This has absolutely nothing to do with intelligence; and is simply a description of statistical AI. It is no coincidence that this "theory" is only even known in that community, let alone popular elsewhere.
You cannot derive theories (explanations, concepts, understanding, representations) by induction (compression) across measurements. There's lots and lots of work on this, and it is beyond reproach -- for a popular introduction see the first three chapters of David Deutch's fabric of reality.
"Compressionists" might say that representations of the target domain are "compressed" with respect to their measures insofar as they take up less space. This is quite a funnily ridiculous thing to say: of course they do. This has nothing to do with the process of compression... it is a feature only of there being an infinite number of ways to measure any given causal property.
The role "compression" here plays, insofar as it is relevant at all, is a trivial an uninformative observation that "explanations are finite, and measures infinite" -- one is not a compression of the other.
A "door" for example isn't a fundamental object of the universe. Its a collection of atoms or quarks or whatever you consider fundamental objects of reality, (or even a completely abstract object having some resemblance in space/time), but each part in itself means nothing. It is the collection of them in a certain configuration which we recognize as being a "door".
It is precisely the "compression" of of bunch of data points into a simpler concept.
The same goes for concepts like shapes, diagonal, alternating, etc. There are endless infinite patterns from which we learn to distinguish
The only sense in which physics is "compressed" wrt measurement data is just that it's conducted using finite sequences of mathematical symbols given a causal semantics.
The only "data" which can be compressed into, eg., newton's universal law requires you already knowing that law in order to collect it. Almost all measurements of the sky need to be adjusted, or ignored, by already knowing the theory.
The vary nature of the inquiry of physics is to find the simplest mental conception which explains the greatest number of physical phenomena. To improve any system or model while retaining all its qualities, it is necessary to deconstruct it into simpler bits so that it can be reconstructed into something simpler.
Indeed, one can conceive of a physics model in which every observable phenomena is a unique entity which has the properties of doing exactly what is being observed. But attempting to find common properties between phenomena, we can reduce them to simpler explanations, and this can happen in iterative steps. In the the primitive studies of physics, this was reducing the elements into "fire", "water", "steam", "earth", etc. Then we broke this model down further, by saying all elements have something in common, in that they are atoms. And we then attempted to explain commonalities by which atoms behave by breaking it down further.
Actually, compression-is-intelligence has been a relatively popular theory for a couple decades. The Hutter prize from 2006 is based around that premise.
The idea is that compression and abstraction are fundamentally the same operation. If you had a perfect compressor that could compress the digits of pi into the 10-line algorithm that created them, in a deep sense it has understood pi and could generate as many more digits as you want.