You're using "information" in a highly colloquial manner here. It's when one gets into the specifics of these different examples, one cannot simply equate information in these different instances. To do so is again equivocating.
People use analogies like language or computer code when talking about DNA because they are trying to make the concept relatable. This doesn't mean that DNA is an actual language or computer program.
It not just any information, it is absolutely a hierarchical digital information system, an entirely objective and definitive phenomena. It is not merely an 'analogy' and Dawkins goes out of his way to make this point below- not saying he is the ultimate authority, but obviously not 'batting for ID'
As he fleshes out- it is a true code, because there is a code convention- i,e, it's not the characters in the code that directly produce the result, they represent actual information which is copied and interpreted for a specific result. As a book is interpreted by a common code convention that the author and the reader share- it's not the 'ink on the page' that actually tells the story, but the information it describes- do you see the distinction?
"After Watson and Crick, we know that genes themselves, within their minute internal structure, are long strings of pure digital information. What is more, they are truly digital, in the full and strong sense of computers and compact disks, not in the weak sense of the nervous system. The genetic code is not a binary code as in computers, nor an eight-level code as in some telephone systems, but a quaternary code, with four symbols. The machine code of the genes is uncannily computerlike. Apart from differences in jargon, the pages of a molecular-biology journal might be interchanged with those of a computer-engineering journal. . . .
Our genetic system, which is the universal system of all life on the planet, is digital to the core. With word-for-word accuracy, you could encode the whole of the New Testament in those parts of the human genome that are at present filled with “junk” DNA – that is, DNA not used, at least in the ordinary way, by the body. Every cell in your body contains the equivalent of forty-six immense data tapes, reeling off digital characters via numerous reading heads working simultaneously. In every cell, these tapes – the chromosomes – contain the same information, but the reading heads in different kinds of cells seek out different parts of the database for their own specialist purposes. . . .
Genes are pure information – information that can be encoded, recoded and decoded, without any degradation or change of meaning. Pure information can be copied and, since it is digital information, the fidelity of the copying can be immense. DNA characters are copied with an accuracy that rivals anything modern engineers can do."