Three different things can be called a download
A movie file, an application installer and a saved-list backup are not interchangeable. Cinego provides a text backup for My List. These pages do not supply movie files or native APK installers.
For a film you want offline
Open the film’s viewing choices and check whether the selected service supports downloads for that title in its application. Offline availability can depend on the plan, device and edition. Downloads may also have expiry or renewal conditions. Use the controls and information shown by that service rather than expecting a browser bookmark to save the film.
For your list
Choose Create backup on My List, copy all of the generated text and keep it in a note. Restore it through Merge backup in the other browser. The contents identify saved titles and episode checkpoints. They contain no video and cannot be played.
For a home-screen icon
A bookmark or supported shortcut simply opens Cinego more conveniently. It does not replace an offline download. See browser and home-screen access, or back up your list before changing devices.