The non-coding genome is a large part of what causes us to look different from a mouse or a fruit-fly. Gene regulation (what gene to transcribe, and when) is largely controlled by non-coding DNA: https://en.wikipedia.org/wiki/Regulation_of_gene_expression
There are energy costs associated with useless information being stored and transcribed in the genome... so it is pretty reasonable to believe that information theory does play some (however minor) role in how data is stored.
The fact that the DNA involved in gene regulation is mostly non-coding does not mean that most non-coding DNA has a regulatory purpose. If it did, changes in non-coding DNA would be more important, in aggregate, than changes in coding DNA are, with respect to evolution and disease, on account of its prevalence.
Most non-coding DNA does have a regulatory purpose. Things like copy number variations (where information is "duplicated") between promoters and enhancers can can change the rate at which particular genes get transcribed. This is my point: that viewing the genome as "information" in the simple sense of each ATCG nucleotide corresponding to bits is absurd. The genome is a geometric AND chemical entity and encodes information using both paradigms.
My point is not to say that you are wrong, but rather that you do not know. It is FAR from well established that the genome is inefficient at storing information.
From an information-theoretic point of view (which is where we started) even if all non-coding DNA had some purpose, it would not be a sufficient basis for assuming that it is close to the most information-theoretically efficient implementation. On the other hand, the fact that whole genomes can be handily compressed, with straightforward Huffman encoding techniques, settles the infomation-theoretic efficiency question, regardless of whether your speculation about the usefulness of all non-coding DNA is correct.
> The non-coding genome is a large part of what causes us to look different from a mouse or a fruit-fly.
Now that's just fantasy. There have been a lot of changes since our common ancestors with mice and fruit flies. We are all animals and obviously would share genes that perform analogous functions -- like common genes for controlling wing shapes and organ shapes. But would you like to point me to evidence in the human DNA that codes for fruit fly wings? And what role do you think having virus DNA in our genome plays in our development?
DNA encodes information. Information theory does play a role. But as mannykannot pointed out, the topic of the conversation is something else.