No. The problem is not with Object, but that you are creating two different objects.
The object literal syntax ({foo: 'bar'}) declares an object inline. When the script is executed, the object is created. Using that syntax multiple times creates multiple objects.
You can easily test that with {foo: 3} === {foo: 3}. This will evaluate to false, but they are not the same object (reference).
The indexOf method checks if the object, string, number, etc, is present in the array. You're passing a new object, which is not in the array.
If you have a reference to the object, you can use that and indexOf will work:
var foo = {aa:5,bb:5}, bar = {aa:6,bb:6};
var array2 = [];
array2.push(foo);
array2.push(bar);
console.log(array2.indexOf(foo));
Because you're referring to the same instance, this will print the index.
You can also use filter or find with a predicate to perform a deep search:
function deepIndexOf(arr, obj) {
  return arr.findIndex(function (cur) {
    return Object.keys(obj).every(function (key) {
      return obj[key] === cur[key];
    });
  });
}
var array2 = [];
array2.push(foo);
array2.push(bar);
console.log(deepIndexOf(array2, foo));
This won't recurse into nested objects, but will accomplish the comparison you're looking for (equivalence on two objects and their immediate fields).