A two part language like this would likely evolve when it is for some reason important to interfere with communication.
Which is kinda counterintuitive, since the point of language is to communicate.
To make it work, in this scenario, there is a spoken language, the one they use to learn these stories/myths, and to speak plainly with each other, there is a sign language likely used for the same reason ours was made (to communicate when speech/sound is impossible or dangerous), there may well be other spoken languages, and/or other sign languages, there are multiple cultures with mythologies, religions, shared stories. They all have to exist as a background for how this two-part language is set up, how it evolves.
This two-part or hybrid language is more difficult to track or understand than either a sign language or a spoken language - one must be looking and listening, one must pay attention to two flows of communication simultaneously, one must intuit how they interact with each other in support, in negation, and in supplement, all at the same time. Not easy, just think about trying to follow three or four conversations at once, and realize losing track of any is likely to mess with understanding the others, as well, since they interweave.
And on top of that, the spoken part of the language is made even harder to understand by generously incorporating references to mythology and stories, culture, etc. This makes it very difficult to follow, only one knowing the right things, paying the right sort of attention, and with enough presence of mind to translate on the fly is going to get, even, most of what the one speaking is trying to say.
In the end, this is set up to be as difficult to understand as possible... and so this kind of language will only evolve if there is a reason to be difficult to understand. A reason like trying to communicate with certain people at the same time/place as needing to not communicate with other certain people, who are there watching. There has to be a reason someone would rather risk misunderstanding or completely miss communicating with their target, just to be sure someone who isn't their target, will not understand.
One reason might be oppression, war, slavery, other similar deadly troubles. A member of an oppressed people may well need to hide their communications, and not only will speaking in metaphor and reference hide depths of meaning, the use of a secondary communication route (the sign language) may let them negate or reverse certain parts of their speech, or emphasize them, or so on, so they can sound like they're saying one thing to those listening, while sneaking a second communication under others' noses. In this case, intercepted communications is a life-or-death matter, so the more layers of misdirection they can come up with, the better.
Another reason, a bit more lightly, might be rivalry between groups, competition, politics, etc. In this case, the references would be different for each group, different mythologies or emphasizing different stories within their mythology, based on which individuals, which stories or lessons their group related to... and the sign language would be different as well, depending on what kinds of things they wanted to reference, and what stories they were weaving their communication into. Intercepted communications may effect status, might spill secrets, might skew interactions between the groups, so it can be quite serious without being deadly.
So, to wrap things up a bit, this double pronged language evolves in a scenario where there are multiple groups, multiple languages, and they are in competition with each other - so this evolution is a way to communicate with members of their own group, and confuse or deny communication with those who are members of different groups. This double language would begin as a pidgin of these two languages, used as a code or encryption, with set call-and-response pairs, to communicate specific things without being overheard, so to speak. Over time, it may evolve into a separate language of its own, a creole or a lingua fraca, even though the base languages would likely also be known, by the same people, for a very long time... by the time this double-language would stand on its own, it would have simplified the mythological references down to the bone, to be easier to understand, so to maintain that extra layer of references the base language must be available for plain speech.