Okay, so two things to hit here. I’ll start on the more important aspect of your topic.
Whitelisting and all doesn’t work like it used to. So if you’re trying to block cookies except for certain sites or something, you’ll need to look toward Forget me when I close this site and all in Shields instead.
So go to Settings → Shields (or just go to brave://settings/shields) and turn the global setting on. This will make it so all websites will delete your data when you leave. Then you just turn off the toggle for it on any site when you visit by hitting the Shields lion and turning off the toggle while you’re on that site. ,
NOTE/WARNING
A lot of people have been trying to use the Clear Browsing Data with On Exit options. Doing this overrides anything else in your settings, meaning it will indeed clear all cookies and all even if it’s on a whitelisted site. So definitely want to avoid this if you don’t want all deleted. Instead just use the Shields option as mentioned just before.
And as to this part, it’s something that changed a while ago in all Chromium browsers (happened in February or March I think). What you’re wanting still exists but they renamed it. What you would now be looking for is On-device site data.
It’s located very similarly to where it used to be. Settings → Privacy and security → Site and Shields Settings → On-device site data (which is down toward the bottom, you might have to click on additional content settings to see it)
Alternatively, you can just type in brave://settings/content/siteData to go straight to it.