Here is a video with the bug in action:
https://watch.screencastify.com/v/qykDYcjYSGiFH8nFXMRN
I’m using:
Version 1.36.109 Chromium: 99.0.4844.51 (Official Build) (x86_64)
Seems like others ran into this as well:
Drag and drop not working in Version 1.35.89 Chromium: 97.0.4692.71 (Official Build) beta (64-bit) (closed)
Hi and welcome to the community. 
I can’t help, sorry. I was wondering what OS you are using? It appears that most (if not all) of the previous posts were related to Linux distributions.
Also, from reading some of the older posts, it appears this was a Chromium issue that impacted Linux users. Maybe it was reintroduced with the new update?
And finally, just so you are aware, sometimes an issue is fixed without user notification. Unfortunate, but it is what it is.
I’m using MacOS Big Sur 
I am on linux and the latest brave browser does not have this issue. I can drag and drop normally.
It seems to be an actual bug affecting entire macOS builds or only limited to OP’s device.
+1 for can’t drag/drop bookmarks
MacOS 12.2.1
Brave
Version 1.36.112 Chromium: 99.0.4844.51 (Official Build) (x86_64)
I CAN create a folder in Bookmarks Bar, and then drag new URLs to it. I can’t drag to imported bookmarks bar folders
@billmeade Hi and welcome to the community!
I’ve linked a more active post below. A Brave GitHub issue report has been opened and they are working on solving. You can view the issue report and track progress on the Brave GitHub site. The link to the report is in the post.
+1 for can’t drag and drop. I have the same issue on Windows 10 and Linuxmint 20.3 MATE 64bit.