I have the same Problem. It shows me that this the program was purchased on another computer and I must enter my apple account Login and Password in order to enable Cookie 5.8.1 on my computer.
Same Problem. After new installing, doing so just gave me the unending request for AppleID and password. I finally gave up and found an older version (5.7.8) on a backup and went through the setup process. This version appears to be working. Based on this, the installation problem appears to be associated with version 5.8.1.
Can not update Cookie 5.8 to Cookie 5.8.1 in AppStore!