It's been a long time since I have cracked a physics book, but your mention of interesting "fundamental physical quantities" triggered the recollection of there being a conservation of information result in quantum mechanics where you can come up with an action whose equations of motion are Schrödinger's equation and the conserved quantity is a probability current. So I wonder to what extent (if any) it might make sense to try to approach these things in terms of the really fundamental quantity of information itself?
Approaching physics from a pure information flow is definitely a current research topic. I suspect we see less popsci treatment of it because almost nobody understands information at all, then trying to apply it to physics that also almost nobody understands is probably at least three or four bridges too far for a popsci treatment, but it's a current and active topic.
This might be insultingly simplistic, but I always thought the phrase "conservation of information" just meant that the time-evolution operator in quantum mechanics was unitary. Unitary mappings are always bijective functions - so it makes intuitive sense to say that all information is preserved. However, it does not follow that this information is useful to actually quantify, like energy or momentum. There is certainly a kind of applied mathematics called "information theory", but I doubt there's any relevance to the term "conservation of information" as it's used in fundamental physics.
The links below lend credibility to my interpretation.
