How to remove a element from array in dropdown of list2 after selecting an element form list1 drop down in Angular js?

You can create a filter to do that instead of using ngChange directives.

First of all, make ngModel useful:

<select ng-model="xaxis" ng-options></select>
<select ng-model="yaxis" ng-options></select>

Second, to filter the selected item in another select and make it disappear in the current select, we'd like to use a filter in ngOptions directive like below:

<select ng-options="l1.columnName for l1 in list1 | onlyNotSelectedCouldShowHere">

The name onlyNotSelectedCouldShowHere is a little tedious, right? So we'll only use notSelected as the name in the real code below. :P

To make it work, we need to create a filter (just appended to myApp):

myApp
  // Create a filter
  .filter('notSelected', function(){
    return function(list, target) {
      // If there is no selected item in xaxis or yaxis, return all of the items 
      if (target === undefined) return list;

      // Use Array.prototype.filter() function [ES5]
      var filtered = list.filter(function(item) {
        return item.columnName !== target.columnName;
      });
      // Return the filtered array. 
      return filtered;
    }
  });

After realising that we need to pass an argument to the filter (yaxis for the x select and xaxis for the y select), we update the html:

<select ng-model="xaxis" ng-options=" l1.columnName for l1 in list1 | notSelected:yaxis"></select>
<select ng-model="yaxis" ng-options=" l2.columnName for l2 in list2 | notSelected:xaxis"></select>

Check out the updated jsfiddle.

To learn more about the filter in AngularJS, I've found an excellent post: Everything about custom filters in AngularJS by Todd Motto.

Update on the early morning of 8/7/16 in China: when value="" is selected, show the full list.

myApp.filter('notSelected', function(){
  return function(list, target) {
    // undefined: If there is no selected item in xaxis or yaxis, 
    // null     : or if value="" is selected
    // return the full list
    if (target === undefined || target === null) return list;
    var filtered = list.filter(function(item) {
      return item.columnName !== target.columnName;
    });
    // console.log(filtered);
    return filtered;
  }
});

Check out the updated jsfiddle.