I’m having the same issue, on Linux Mint 20.2, Cinnamon Edition. So, I don’t think it’s going to have much to do with the running desktop manager.
Tabs drag off from their windows, and cannot be put back, if you try to drag a link or url to the top to open a new tab, it simply forwards the event to the desktop manager, so I ended up with a bunch of text files sitting on my desktop, instead of new tabs being opened.
And this didn’t happen until this last update, 1.34.80 (according to apt), “1.34.80 Chromium: 97.0.4692.71 (Official Build) (64-bit)” according to the brave://version page.
I’m going to try to roll back to an earlier version, if I can, via Synaptic, see if that reverts this behavior.
Edit: NOPE, can’t roll back to the build immediately before this one, only options I get via Synaptic are 1.34.80 (current), 1.33.106, 1.32.115, 1.32.113, 1.32.106, and 1.31.91. Shucks.
Edit, the 2nd: Well, I got proper tab functionality back, but it meant I had to disable the “Use system title bar and borders” setting. Now I can drag tabs left and right, I can drag tabs out of a window and back in, etc.
HOWEVER, this functionality should still have worked, REGARDLESS of that setting.
Now, if I might make a suggestion, for future updates? Most software vendors/devs also provide a copy of (or a link to) the current changelog via apt (or rpm, etc), so users can see what changes have been made, and can probably better assist in troubleshooting, when issues like this arise.