DPoisson. DFish!
I'm primarily a duck call maker, so I can speak to that better than most other calls.
You're right that most sound we hear is not a 'pure tone' like what you'd hear off of a tuning fork. Even the sound coming from a guitar or other instrument is not a pure tone. Same goes for our voice or the voice of an animal- it is a complex tone made of many notes.
With duck calls, there are definitely some out there that get a very basic sound, and as a result, it just does not sound like a duck. You'll hear a lot of makers talk about 'ringing' hail calls, 'raspy' bottom ends or even 'cajun' squeals in their calls. These are all effectively overtones that add to the complexity of the note that the call is making while operated to make it sound even more like a duck and the complexity of their voice.