Hello!

When I exit the Opera browser, my saved site login data is deleted and I have to log in again, but after closing the browser, the story repeats.

The solution is that the browser cannot access the secure keystore on your Mac. In this solution, we will delete all existing stores of the Opera browser and restart the browser, after that a new store will be automatically created and it will already work correctly, and site data will begin to be saved.

Step 1.

Open «Keychain Access» app on your macOS.

Step 2.

Select «login».

Step 3.

Select «Passwords» item.

Step 4.

Find «Opera ...» in the list.

Step 5.

Remove any matches found.

This is all! :)

Now you can close the «Keychain Access» application window and also restart your Opera browser. 
After that, try again to save the logins in the Opera browser.
Now it will definitely work :)