Previously, Brave did never save any website login passwords in one user account.
Safari did save logins for that user account.
When making a backup recently, the backup app logged an error when backing up the keychain file of that user account.
I deleted the keychain. MacOS automatically generated a new keychain file and now Brave saves all passwords and I am always automatically logged in on all websites.
This might be a fix for most or all users.
The issue has been tracked here