No mention of birdsong in this article leaves me hanging. Surely birdsong has got to be much more studied, and must contain lots of complexity ripe to be plumbed for hidden meaning.
IIRC there have been studies and they showed differences in the latter depending on the specific threat. But "birds" is a large group and studying "bird songs" is not any more helpful than studying "mammal noises" and I don't recall which species the study looked into.
It's worth remembering though that "language" is a form of communication and communication is vital in all social species. "Come here, I'm a good mate and available" and "Go away, I'm big and scary" are the lowest common denominator so even non-social species usually have some way of expressing those sentiments. But the more complex a species' social dynamics are, the more likely it is they will have means of communication capable of reflecting that.
It's also a mistake to look at utterances (mouth sounds) in isolation, especially when looking at a communication system we're unfamiliar with. Body language and facial expressions are a big deal in human communication even if we sometimes exclude them from analysis for simplicity. Not to mention that language features may be different if only because of the different biology (e.g. overtones, repetition or rhythm might carry semantics).
In other words, what Arrival did with alien "written language" equally applies to non-human "spoken language". It would be a mistake to assume that all the concepts we use for analyzing human language would apply to forms of communication in other animals even if another animal's language might turn out to be similarly complex.
