<p>A strict equality operator will tell you if two object <strong>types</strong> are equal. However, is there a way to tell if two objects are equal, <strong>much like the hash code</strong> value in Java?</p>

<p>Stack Overflow question <em><a href="https://stackoverflow.com/questions/194846">Is there any kind of hashCode function in JavaScript?</a></em> is similar to this question, but requires a more academic answer. The scenario above demonstrates why it would be necessary to have one, and I'm wondering if there is any <strong>equivalent solution</strong>.</p>

Depends on what you mean by equality. And therefore it is up to you, as the developer of the classes, to define their equality.
There's one case used sometimes, where two instances are considered 'equal' if they point to the same location in memory, but that is not always what you want. For instance, if I have a Person class, I might want to consider two Person objects 'equal' if they have the same Last Name, First Name, and Social Security Number (even if they point to different locations in memory).
On the other hand, we can't simply say that two objects are equal if the value of each of their members is the same, since, sometimes, you don't want that. In other words, for each class, it's up to the class developer to define what members make up the objects 'identity' and develop a proper equality operator (be it via overloading the == operator or an Equals method).
Saying that two objects are equal if they have the same hash is one way out. However you then have to wonder how the hash is calculated for each instance. Going back to the Person example above, we could use this system if the hash was calculated by looking at the values of the First Name, Last Name, and Social Security Number fields. On top of that, we are then relying on the quality of the hashing method (that's a huge topic on its own, but suffice it to say that not all hashes are created equal, and bad hashing methods can lead to
more
collisions, which in this case would return false matches).