Andrew Huang does a good job of explaining harmonics and overtones[0]. You only need to watch until about the 4 minute mark (from the timestamp) to get an explanation of harmonics and what the sonograms represent.
The short of it is that most natural sounds product a root tone plus a varying amount of related tones above it. Our ears hear the root tone, and the other tones above it are what give the sound its uniqueness. That's why a guitar, a clarinet and 3 singers can produce the same note while sounding distinct.
Birds seem to produce a natural sound without a lot of the related tones above it. Their sound is, relatively speaking, much purer than most other natural sounds. That's very unique.
The short of it is that most natural sounds product a root tone plus a varying amount of related tones above it. Our ears hear the root tone, and the other tones above it are what give the sound its uniqueness. That's why a guitar, a clarinet and 3 singers can produce the same note while sounding distinct.
Birds seem to produce a natural sound without a lot of the related tones above it. Their sound is, relatively speaking, much purer than most other natural sounds. That's very unique.
[0]: https://youtu.be/Wx_kugSemfY?t=95