I think some clarification may be required. You may think "Mate" is one entity. A so-called "desktop environment". It's not. It's a collection of various applications and components, including a window manager, that makes up what "Mate" is. Each of those components are 'clients' from the perspective of Xorg. It's the Xorg server that actually shows you the screen. The good ol' X11 protocol was made so it was possible for the application (the X client) to run on an entirely different machine from where the monitor and screen (the X server) actually was.
Nowadays it's obviously more common to have your application and (graphical) screen on the same computer. So this whole "remote" thing of X11 kind of gets in the way. That's the 'problem' Wayland, for example, tries to solve.