var arr = [];

for (var prop in obj) {
   if (obj.hasOwnProperty(prop)) {
      var innerObj = {};
      innerObj[prop] = obj[prop];
      arr.push(innerObj)
   }
}
    
console.log(arr);

here is demo https://plnkr.co/edit/9PxisCVrhxlurHJYyeIB?p=preview

There are no keys in JavaScript arrays. Use objects for that purpose.

var obj = {};

$.getJSON("displayjson.php",function (data) {
    $.each(data.news, function (i, news) {
        obj[news.title] = news.link;
    });                      
});

// later:
$.each(obj, function (index, value) {
    alert( index + ' : ' + value );
});

In JavaScript, objects fulfill the role of associative arrays. Be aware that objects do not have a defined "sort order" when iterating them (see below).

However, In your case it is not really clear to me why you transfer data from the original object (data.news) at all. Why do you not simply pass a reference to that object around?

You can combine objects and arrays to achieve predictable iteration and key/value behavior:

var arr = [];

$.getJSON("displayjson.php",function (data) {
    $.each(data.news, function (i, news) {
        arr.push({
            title: news.title, 
            link:  news.link
        });
    });                      
});

// later:
$.each(arr, function (index, value) {
    alert( value.title + ' : ' + value.link );
});

You have to create an object. Assign the values to the object. Then push it into the array:

var nietos = [];
var obj = {};
obj["01"] = nieto.label;
obj["02"] = nieto.value;
nietos.push(obj);