Currently, Amazon Music not providing any official API to access to an user Amazon Music account data.
While there's no public API and we don't found a secure workaround to access to Amazon Music user datas, we are not able to move datas TO Amazon Music.
But we are still able to transfer an Amazon Music playlist to other services by using the "share link" of a playlist.
You can see a thread about this request to Amazon here => Forum Amazon Developer
Help Soundiiz by writing on Twitter to @AmazonMusic and asking to open an API for Amazon Music.