Thank you kzhang, I wouldn’t have figured it out without your comment.
In my case (ArchLinux + XMonad, installed via the obsidian AUR package) I had to change frame: false to frame: true in two places in the file for the window border to be visible and controllable by my window manager.
In case it may help someone, I’ll post my process as well.
cd /usr/lib/obsidian
sudo npx azar extract obsidian.azar obsidian.azar.extracted
sudo vim obsidian.azar.extracted/main.js # edit and save
sudo npx azar pack obsidian.azar.extracted/ obsidian.azar
and then restart Obsidian, and the border will be visible.