But if I refer to another page and click on it, they change the url at address bar, but it doesn't update the page to read and show the referred one. Even though I copy and paste the url with another permalink (another page) at the same window, it doesn't update the page, I need to create another browser window/tab.

I think it helps: OnHashChange

What do you think of this:

window.addEventListener("hashchange", function(e)
  {
     var parts = window.location.hash.match(/[^\/]+\/?/g);
    parts.shift();
    parts.forEach(function(part) {
      socket.emit('readFile', {name: part});
      $('#navigation').children().attr('class', 'link');
      $('#navigation a:contains("'+part+'")').attr('class', 'selected link');
    });
  }, false);

It's a repetition of emission of hash used on connect event. I think you can embed this routine as a function, since it's a repetition. Oh, and we need to highlight the proper link at the menu.