I could find some physical interpretations of information for quantum systems, but not for classical physics. Can some interpretation of information for macroscopic phenomena given? Say, the electronic interpretation, when a bunch of photons/electrons/ions deliver a bit of information, can it be related to the "bit of physical information"?
Yes, definitely. As in Wolphram jonny's link on information theory, there is definitely a notion of information in classical physics. If we have a single classical two-state system which has even odds of being in either state, then it takes exactly one bit of information to fully specify which of the two states it is in. Since real systems are never exactly two-state and we never exactly know which state they're in, we never have a precise notion of information in real systems, but we have very good approximate notions.
One way to describe uncertainty in classical physics is to give the probability density that the system is in a certain range of states specified, in the Hamiltonian formalism, by positions and conjugate momenta. Liouville's theorem tells us that this phase space density is conserved by Hamiltonian dynamics, meaning that since classical physics is deterministic, it can't destroy (or create) information. However, random dynamics can destroy information, and even deterministic dynamics can render information inaccessible by spreading it into the wider world (dissipation) or into microscopic degrees of freedom (chaos).