Obsidian seem to rely on raw key code rather than the assigned key output.
If I press the key that was initially assigned ` Obsidian switches the windows.
This is also confirmed by the fact that a different method of key rebinding works. Specifically, Karabiner is a virtual keyboard driver. It intercepts keycodes and remaps them before the OS processes them. With this method Obsidian properly responds to the shortcut.