I have got the same problem. I just got my new kindle fire 7, 2017 today. I installed spotify and it doesn't get any further than a black screen. I have unstalled it several times. I even sideloaded a version from apk mirror just to see if it was the version on the amazone app store that is faulty. My feeling is that it has to do with the latest update to fire OS. I have a fire hd 6, 2014 and spotify is working fine. Hope they fix it soon. I am currently residing in Switzerland.
what i did was, disconnect the device from the W-Fi, the open the app; it open this time with no issues, reconnect to Wi-Fi and finish the log in process.. close the app and then it started opening fine with no issues.. not sure what was the issue but right now it is opening without any issues..