JavaScript modular framework for a large web-site
Very readable code, some minor comments:


Some folks dislike the lacking curly braces for one liners, I dont mind it here myself
JsHint will complain about you creating a function in a for loop, I think you should indeed refactor that
Using this[name] to register modules would be to fickle for me (shenanigans with bind, apply, and call come to mind), I would prefer a self executing function with a var statement. 
You only have 1 line of comment, for a very straightforward piece of code, I would have commented the moduleExports[exportFnName] = (function (exportFn) { line
I would split this out in to 2 different error messages:  if (!name || !module.init)
        throw new Error('Could not load a module ' + name);
Actually, in your code I see that init is a bit contrived, I would not make that mandatory personally


Now, when I take a step back, I think you might be suffering from Second System Syndrome. Consider that in effect you are replacing

$("button").click(function() {
    $('input').attr('data-color'));
});


with

kibfw.register('decorator', {
  init: function () {
    this.$decorateItems = this.$('input');
  },
  context: {
    $decorateItems: undefined
  },
  exports: {
    setColor: function (color) {
      this.$decorateItems.css('color', color);
    }
  }
});

// Usage:
$("button").click(function() {
    kibfw.decorator.setColor($(this).attr('data-color'));
});


I like window.kibfw, it is pretty beautiful. But I would hate to have to write a website with it.