Note that you can also use emacs as your window manager, if this is running in Xorg, so your emacs window can be in emacs too.
https://github.com/emacs-exwm/exwm
EXWM (Emacs X Window Manager) is a full-featured tiling X window manager for Emacs built on top of XELB.
There's some work on a Wayland compositor, but that's not as mature.
https://codeberg.org/ezemtsov/ewm
EWM is a modern, fast Linux desktop environment based on Emacs. It is written in Rust, built on Smithay, and runs as a dynamic module inside an Emacs session. Wayland applications appear as Emacs buffers, which means graphical windows can be managed with the same commands, keybindings, and workflows as the rest of Emacs.
The ultimate goal of EWM is to let power users be bottlenecked by their own thinking, not by the tools they use. Computers should never stand in our way. EWM is built around the idea that window management, application launching, navigation, text input, and system control should be fast, deterministic, programmable, and composable.
EDIT: I guess technically you could use Wayland, EWM making emacs the compositor, then use xwayland in that, with EXWM making emacs the window manager in that, in which you have an instance of X11 emacs running which itself is running...
EDIT2: No, wait. I don't know if you can run a window manager directly in xwayland.
considers
You could run it in Xephyr in xwayland, though, and you can run a window manager in that. So Wayland->emacs(EWM)->xwayland->Xephyr->emacs(EXWM)->emacs->emacs->emacs...
