Solution 1:

Unfortunately for this menu it is only possible to do manually with the dconf-editor.

  1. install dconf-editor sudo apt install dconf-editor
  2. go to org > mate > panel > menubar
  3. change default icon-name to your new icon name

if you want to have a custom icon, copy the .svg image to /usr/share/icons/apps/image.svg replace image with the image title of course...

https://ubuntu-mate.community/t/how-i-can-change-main-menu-icon/9884/3

PS: Using Manjaro you simply have to put your prefered icon to ~/.icons/logo-manjaro.svg

PSS: I found the name of the menu's icon using Gtk Inspector

Solution 2:

Use gsettings command for a one-line solution.

To change the icon named foo:

gsettings set org.mate.panel.menubar icon-name 'foo'

The string within single quotes refers to the icon name of any theme that has been installed properly in /usr/share/icons and ~/.local/share/icons directories. To find out icon names that are available by default, refer to the Icon Naming Specification.

For example, replace 'string' with 'user-bookmarks' to show the bookmark icon on the main menu. The bookmark icon may appear as a yellow star, a red heart, or a folder with star emboss, depending on which icon theme is used.

Unfortunately, MATE main menu has no easy way to use a custom icon without changing the system-wide theme. The MATE main menu will look for the icon name of currently used theme and that makes difficult to use a single custom icon.

To restore the icon to default:

gsettings reset org.mate.panel.menubar icon-name

Note that the question is specific to the MATE main menu applet. A different menu applet--the advanced MATE menu--uses a different schema, or even better, already have an option to change the default icon from the preferences.

Tested in Ubuntu MATE 18.04 (MATE 1.20)