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.

https://en.wikipedia.org/wiki/Time_evolution#In_quantum_mech...

https://en.wikipedia.org/wiki/Bijection

https://en.wikipedia.org/wiki/Black_hole_information_paradox