Why can't the plugin overwrite a file or create a backup? What are the necessary permissions?

The Unix systems (this includes Linux in all its flavors and distributions) have permissions and ownership for each file (read more here).

We have often seen that when a website is moved from one hosting provider to another that uses Linux (most of the market), the permissions are correct but the ownership of a file is not correctly preserved, or vice versa. For example, a file will have the correct write permissions, but the file owner will be guest instead of www-data, meaning that if you use a plugin such as Enable Media Replace or ShortPixel Image Optimizer, your attempt to replace a file will fail.

Contact your hosting provider after moving a website from one host to another and make sure that all your files have both the correct ownership and permissions. Once this is sorted out, you should be able to replace files using plugins like the ones mentioned above.