I see shell as an instrument through which you use other tools. In that sense, vi feels much more like a shell, because you have to use other standard unix programs with `:.!` for much functionality.
Where Emacs comes with all bells and whistles included in one big distribution, much like an operating system.
Most people aren't using vim that way. For your example, tmux would be more like emacs in the sense that you live in it, manages your windows, etc. Vim is just another tool in some window along with others.
Something that undermines the operating system angle is that emacs does not implement hardware drivers. All interaction with the computer go via system calls.
Norton Commander contains a text editor. Emacs operates at that level, whilst being reprogrammable.