Does it? I don't see that claim being made anywhere.

They do say that the audio is processed locally, but that does not preclude them from making an API call to find a signature match.

> The audio stream is processed locally on your device and never recorded.

I just tested and a song that works fine normally failed as soon as I turned Airplane Mode on.

ShazamKit is not third party, it is provided by Apple. Their website states "The audio stream is processed locally on your device and never recorded.", though this could refer to the audio statistics being collected locally, I assume ShazamKit still contacts an Apple API to get the classification.