As in, to have the browser clear cookies on exit except for specific websites.
In a way this is also for myself as once in a while I look for the setting in a wrong place and a handful of outdated forum posts aren't helping.
The setting
In Vivaldi, you can set the browser to clear cookies on exit. This is done in Settings ➜ Privacy & Security ➜ Website Permissions ➜ Global Permissions
Adding exceptions "on spot"
When you are on a website, you can click on the "lock" icon, click "Cookies and site data", click "Manage on-device site data", click on the vertical ellipsis ⋮ button, and click "Allow to save data"
This will add an exception for the website's domain and its cookies will no longer be deleted on browser exit despite the "Session Only" preference.
Adding exceptions in Settings
You can also add/view exceptions in the aforementioned Settings section.
To add a website, you would want to click the "+" button, enter a domain name, and then override the desired permissions (in this case, Cookies):
And that's it!