AHA! I just tried the following Drag-n-Drop example to test Brave and it FAILED to do anything. I then tested the same example with Chromium and it WORKED as expected, moving the w3schools image from the left box to the right box:
https://www.w3schools.com/HTML/html5_draganddrop.asp
If someone else who is seeing this problem on Linux could please verify if Drag-n-Drop is working or not on this example or any other web pages, it might be helpful in showing this is a more widespread problem. Thanks.
EDIT: Oops, my installed Chromium is old. I just read that the problem is in Chromium source code used by Brave [1.34.80 Chromium: 97.0.4692.71 (Official Build) (64-bit)]: Moving tabs with click and drag is not working anymore - #4 by Waisrond