It's just like ink on paper. Sure there are physical itneractions between the ink and the paper, and this is why the molecules stay there etc. But the disposition of the molecules were arbitrary, and if a given disposition (a letter) carries any more information then another (a scribble) is strictly because we have all established an arbitrary code in which we decide that such a pattern means such and such, and that other pattern means nothing.
It's not like ink on paper. You could - at a push - perhaps argue it's like electrons in a processor. Ink on paper produces no direct effects anywhere; photons bounce off it and we, eventually, decode those into meaning. DNA interacts chemically and physically with proteins and RNA.
Coded information only exists if their is a semantic aspect to it, without any code it has no information at all.
Thinking of genetics as information, especially information without a strict formal definition, is rarely useful.
It is the same thing with DNA. Somewhere along the line from none-life to life, a code was established either via randomness, via an as-of-yet-unknown natural process, or via an intelligent being. But it wasn't because of any particular physical interaction.
With the proteins it seems to be different. It has information strictly because 'the key physically fits the hole', and this information comes from a real physical basis.
No, it's not different. The DNA interacts with other molecules. That interaction produces changes in the surrounding environment. It's physical all the way down. The encoding in DNA is semantically flexible, as we analyse it, but in the cell the exact same chemical processes that drive protein-protein, protein-lipid or protein-ion interactions are involved in protein-DNA interactions.
No special explanation is required for DNA interactions that is not required for protein-protein interactions.