The term Display Manager is a vestige of the use of X11.

X11 is a Server/Client protocol.

When a user logs in to an XServer, they are given an Xsession. The user can use that Xsession to create one or more X11 Displays (they are just IDs). The X11 Display ID is passed to the X11 client application (that’s what the XDISPLAY environment variable is for). The client apps render their content to that Display ID. This whole thing allows for more than one user to be able to use a single operating system on a single XServer at the same time.

All of that is pretty cumbersome for a user to do themselves in their terminal, that’s what Display Managers are for. They:

• Start the XServer if it isn’t started yet
• Provide a method (eg, login with username and password), to start a new XSession.
• Use that XSession to create an empty X11 Display.
• Look up which is your configured default DE or WM
• Launch the DE or WM with the right parameters, passing it the new XSession and XDisplay

If you’re using Wayland, then the architecture is very different. The Display Manager then simply operates as a login screen.