If you've added songs, albums, etc to your library in the Amazon Music app, you can then search from them in the Alexa app. Amazon Music needs to be your default music service. In the Alexa app, go to Music & Books -> My Music Library and search. It'll bring up matching items from your library.
If you're trying this out be aware it can take a good few minutes from the songs being added to your library before Alexa is aware of them.