Conservation of information is a term with a short history. Biologist Peter Medawar used it in the 1980s to refer to mathematical and computational systems that are limited to producing logical consequences from a given set of axioms or starting points, and thus can create no novel information (everything in the consequences is already implicit in the starting points). His use of the term is the first that I know, though the idea he captured with it is much older. Note that he called it the "Law of Conservation of Information" (see his The Limits of Science, 1984).
The two notions that you refer to in your post are not necessarily the same: The "conservation of information" that is used as a principle in theoretical physics, particular in gedankenexperiments involving black holes, is more commonly known as unitarity. Unitarity became important with the advent of quantum mechanics around 1925 and was investigated rigorously in the late 1920's to early 1930's, resulting in e.g. Stone's theorem on one-parameter unitary groups.
It seems to me that this has very little to do with the "conservation of information" that your second linked article refers to, which is indeed a relatively new concept in (evolutionary) biology, it seems.
Conservation of information is a term with a short history.
This is more or less true and it's good to see that someone has actually come out and said this given the hype surrounding 'information' these days. Until someone comes out and says what actually constitutes information - physically speaking - its hard to say how and why it is conserved; in fact, one could rephrase the above and say:
Information is a term with a short history
This does not mean that the notion of 'information' has no meaning at all to it; in physics, for example, it's a concept that is heavily promoted by Susskind and he does actually give a remarkable argument for black hole entropy based on this which is outlined in his popular book, The Black Hole Wars; given that we are discussing entropy, it is perhaps not surprising the notion of information has turned up here - nevertheless, here, it a measure of all the possible microstates consistent with the macrostate.
The notion of 'information' shouldn't be confused with other notions that appear to deal with 'information' in some not quite precisely defined ways; Unitarity, for example, is conservation of probability and not of information.