Maybe it is just using the filesystem as the database. For example, if it knows all the files it downloads are to a specific folder, then it just scans that folder for previously downloaded files.
Edit: After reading that this application also stores IP Addresses, then either this information must be stored locally or remotely in some fashion. If you are certain it is not stored locally, then the only answer is that this information is retrieved from the server using a unique client identifier such as a session cookie, ip address, or a unique id baked into the application client.