Just my opinion.
It really depends how you use the term (of course). Like, I have no problem if anyone says that oil is a “hydrophobic” substance as long as we don’t extend the usage beyond the implied metaphor. From watching the video, the usage for “code” in question is “a system of signs or symbols”. As @Jon-Perry also points out, the crux falls really under how the terms ‘signs’ and ‘symbols’ are used. I think most people would think that a ‘sign’ are abstract concepts. If that’s how we use ‘signs’ then a codon is not a ‘sign’ since it is not abstract.
Also the rules that describe how the codons are assigned to the amino acids (or as stop codons) is not abstract, nor arbitrary either. There are clear regularities in the genetic code, which gives us clues to how the genetic code originated (see my commentary here). The first comment here by @Dan_Eastwood also mentions an interesting recent paper on this subject.
But we allow ‘signs’ to include physical things that are representatives of something else besides itself, then I find no issue in calling the genetic code a real code. But we can also say that footprints are a code too. They are ‘signs’ that represents a lot of things: What type of animal left the foot print behind, how much they weighed, and in what direction they went. But I really like the example Fraunhofer lines that @Dan_Eastwood gives, since it is reminiscent of bar codes.
This also deflates any argument from analogy (e.g. watchmaker) that ID/creationists often use to claim that the genetic code must be designed since all other codes are designed. That’s why many online atheists become so upset when anyone calling the genetic code a (real) code. They have opted to the position that the genetic code is not actually a code, but I take a more nuanced position.