Intuitively, that’s how I assumed it already worked, and it’s probably what most new users will also expect. But a single global identity also runs counter to the idea of decentralization, and could invite other further complications.
The current experience can still be pretty jarring.
Gnome and some KDE applications used to offer tear-off menus that could be positioned anywhere, similar to NeXTSTEP and its derivatives. Seemed like a neat idea that still deserves some consideration.