Tiling windows

I noticed when hovering over the green button on a window, I have an option to either "move to" or "tile to" the left or the right. When moving one window to either side, I select another window to move it to the opposing side but the "move to" option is no longer available. I don't like using the tile option as it goes into full screen and I can't access a 3rd window without it creating a new desktop.

What alternatives are there besides using 3rd party apps? This is a work computer so we don't have apple id's on them.