You might be interested in this paper:
https://www.researchgate.net/publication/220433962_Concepts_and_Semantic_Relations_in_Information_Science
I've also had luck following its citations and those citing it.
At first the classification looks like a basic semiotic triangle, but it goes into the numerous methodologies for classifying concepts, defining synonymity, etc. via different epistemologies.
I'm not exactly sure how you would model this purely mathematically, perhaps you can't. However, I have a pretty good idea how this could be networked in a SQL database and how one might connect it via DAX or something similar.
You could have, for each word, connected tables with lists of intension and extension, then you'd also need the words grouped into hierarchies in something like the arrangement you tend build for an OLAP analysis server. But of course, you'd need to do second and third order pairing because "three sided shape," would be equivalent to "triangle." Then you'd also want AND and OR relations logged in different tables.
Producing such a database, and getting it to run well with millions of text lookups and many to many relationships would be another matter. My guess is I don't understand the right tools here. You'd probably want to incorporate machine learning and include probability values for words following one another somehow, as well as a database of known contradictions (e.g., "a four sided triangle).
For the original example, you can think of the brain as just such a database, relating different sensory inputs and internal permutations of thought to each other.
I'm not sure if your example contradicts information being physical however. I think the large distances and variations in type might be confusing things.
I could have a computer database that lets me upload pictures to it. I can't speak the language of the person I need to talk to, so I upload a picture of a helicopter. Visual recognition software is already good at this sort of things, it recognizes "helicopter," from my photo and send it along. However, my interlocutor is blind, and speaks Arabic, so the database has to flip the visual representation over to one in sound, in the appropriate language, something computers can already do. If we need to transmit the message back into text, a visual medium, we can do so as well, all within a single computer system.
If information is some sort of non-physical being, how is it that every step of the transformation can be written out as code enacting physical changes in transistors? Brains aren't well understood and make things confusing, but microprocessors are well understood and can do the same things being done in your example. Or are they working with concepts as some sort of Chinese Room? Perhaps. Obviously they don't have subjective experience of the concepts, but that is the only difference apparent in the transformations of information through various mediums of storage that I can see.
Information science tends to focus more on electronic communications. I think there is something to the fact that the entropy of a message in terms of how many meanings it can have is less than the total Shannon Entropy due to synonyms (just made a thread on this point in this same section). Concepts aren't easy to define so they get ignored.
I use a Borges story as a point of reference there and I think another works here, "Funtes and His Memory." The basic plot point is a guy with perfect memory. He can spend 24 hours remembering a day exactly as it happened, fully reliving it. He grows frustrated with decimal systems and just wants to refer to whole numbers by random names, so for example, 7,891 is "Napoleon Bonaparte." The idea being, once he gives a number a name, he never forgets it. He has the ability for perfect extension in definitions. Why talk of dogs when you can refer perfectly to THAT dog, or THAT dog of THAT specific moment?
Unfortunately, Borges doesn't get into the role of universals in compressing information for communication or for predicting the future from imperfect information, where even for Funtes, universals would be useful.
Concepts are necissarily wide nets for groupings of different objects, that is from whence they derive their usefulness. Their ability to be sent via numerous different codes has to do with the fact that they reduce all the information about a particular, to a bite sized amount of information that can easily be coded and transmitted.
The role of concepts in cognition is a bit more interesting, since they help construct subjective experience, but that is neither here nor there.
As to mathematics being known a priori, I would follow Quine on being skeptical on this. The definition of natural numbers requires a circular definition of zero. Parallel lines never met, an a priori fact, until non-euclidean geometries emerged. It seems just as likely that natural selection primes us to understand mathematical relations that reflect the physical world well (indeed, our brains would be based of these same mathematical relations), then that these relations are somehow existent outside their instantiation. Abstract mathematics has developed all sorts of mathematics that don't correspond to physical reality.