At the URL opera://settings/clearBrowserData, make sure nothing is checked on the "On exit" tab.

At the URL opera://settings/content/siteData, make sure the default behavior is set to "Allow sites to save data on your device" and make sure the customized behavior lists are empty.

At the URL opera://settings/cookies, make sure the allow list is empty.

With Opera fully closed, you can check the permissions of the "com.operasoftware.Opera" folder in "/Users/yourusername/Library/Application Support" to make sure it and all its files and folders are not set to read only (have both read and write permissions). Instead of checking, you can open up a terminal, type:

chmod -R u+rw ~/Library/Application\ Support/com.operasoftware.Opera

and press enter to fix any read and write permissions for Opera's user data folder.