I'm having this exact same problem. I've freshly installed OperaGX on my work macbook and it does not save any cookies/session data from any site. My logins to the sidebar messengers are saved as well as my settings, however it's not offering to save passwords and when I quit and reopen opera, I am logged out of all sites and have to log out again. This is even extending to the browser itself; I have an opera account with sync and when I quit and reopen it I'm signed out of opera sync and have to sign back in. if it's relevant, I've also noticed that i can't sign in/confirm my identity to enable password sync/encryption. It opens the tab to ask me to enter my password again and once I enter it, takes me right back to the page asking me to confirm my identity.
I've already reinstalled opera and deleted the profile folders and etc and it has not helped.