Adding elements to object

Your element is not an array, however your cart needs to be an array in order to support many element objects. Code example:

var element = {}, cart = [];
element.id = id;
element.quantity = quantity;
cart.push(element);

If you want cart to be an array of objects in the form { element: { id: 10, quantity: 1} } then perform:

var element = {}, cart = [];
element.id = id;
element.quantity = quantity;
cart.push({element: element});

JSON.stringify() was mentioned as a concern in the comment:

>> JSON.stringify([{a: 1}, {a: 2}]) 
      "[{"a":1},{"a":2}]" 

With that row

var element = {};

you define element to be a plain object. The native JavaScript object has no push() method. To add new items to a plain object use this syntax:

element[ yourKey ] = yourValue;

On the other hand you could define element as an array using

var element = [];

Then you can add elements using push().

To append to an object use Object.assign

var ElementList ={}

function addElement (ElementList, element) {
    let newList = Object.assign(ElementList, element)
    return newList
}
console.log(ElementList)

Output:

{"element":{"id":10,"quantity":1},"element":{"id":11,"quantity":2}}