A sequence of code (signals or text) is not a language. A language is a set of rules for creating and interpreting (meaningful) expressions of information which expressions are composed of basic coding elements that denote basic concepts. The rules for expressing and interpreting information in natural language code can only be derived from examples of text or spoken stories. Only artificial languages have explicit definitions in rules and a set of basic coding elements for concepts. The codons in DNA are the basic coding elements and they denote amino acids as basic concepts. The expressed information is information about how to build functional proteins and higher level functional assemblies. Most likely there are rules for recognizing or creating new high quality DNA code that determine the fabrication of functioning proteins etc. Similar to natural languages, such rules for a bio-language can only be derived from examples of DNA-code. We should take into account that DNA-code not only denote amino acids or determine their sequence in proteins, but also determine how molecular machines and organisms are build. We know the codons, thus, if we would know those rules, then we would know for a newly designed functional protein or molecular machine how its construction should be encoded in DNA code. And more, we should be able to read and write fabrication instructions for 'body plans' for bacteria, plants, animal- and human bodies (provided the ribosomes would accept our code).
A functional bio-language system requires not only rules and code, but also a producer of new code (that can copy or invent by applying the rules) and an interpreter that can also apply the rules and that can do meaningful things with it. In the bio-language case the interpreting machines are the ribosomes, which are like robots that can use the code as instruction for building components (such as tRNA and proteins) as well as machines and organisms that are guided to become assembled from those components.
This bio-language system has similarities with language systems that guide complexes of robots, but they are more than guiding robots to make assemblies, because they also guide the building of the robots themselves and they guide the robots that copy themselves. The only missing component is organisms that invent new machines and code their body plans.
My conclusion: DNA is not a language, but DNA code is written in a bio-language.