A have found the answer to my question.
In GUIDE created and subsequently migrated to Appdesigner dropdown menu, the ValueChangedFcn callback reacts to change of value as well as to click (Changing the name from mydropdown_menu_callback to mydropdown_menuValueChanged did not help). The following code prevented calling twice my callback:
% Value changed function: MatrixM_menu
function mydropdown_menu_callback(app, event)
val = app.mydropdown_menu.Value; % The word in the drop down that is showing.
choices = app.mydropdown_menu.Items; % List of all the possible choices.
[~, selectedIndex] = ismember(val, choices); % The index of the selected item.