Adding items to an object through the .push() method

.push() is a method of the Built-in Array Object

It is not related to jQuery in any way.

You are defining a literal Object with

// Object
var stuff = {};

You can define a literal Array like this

// Array
var stuff = [];

then

stuff.push(element);

Arrays actually get their bracket syntax stuff[index] inherited from their parent, the Object. This is why you are able to use it the way you are in your first example.

This is often used for effortless reflection for dynamically accessing properties

stuff = {}; // Object

stuff['prop'] = 'value'; // assign property of an 
                         // Object via bracket syntax

stuff.prop === stuff['prop']; // true