Where does music file from documents go when airdropped to connected devices?
AirDrop files are stored in the app that matches them. 

For example, pictures will be saved in your Photos app. A presentation will be saved in Keynote. Contacts will be saved in Contacts. And so on.

After you've sent or received an AirDrop file, check the app that matches the file type to ensure it went to the right place and was saved. It should automatically save and open in that app's files.