wp_register_sidebar_widget() disappered my Widgets submenu – what am I doing wrong?

There are some problems in your code but you are close, some pointers to solve problems.

First, eliminate as much code as you can which still contains the problem. In your case remove the second widget, the less code you have to look at the easier it will be to find the problem.

In other words use The SSCCE : http://sscce.org/

Another piece of advice is to create unique widget id’s and prefix your functions, using “Primary” and names like theme_widgets_init is not a great idea.

If you do not see any Widgets available under Appearance in the menu, if the menu item does not exist at all, it is because you need at least one active sidebar using register_sidebar.

You need at least one of these :

 function islam_widgets_sidebar() {
    // Area 1, located at the top of the sidebar.
    register_sidebar( array(
        'name' => __( 'Primary Widget Area'),
        'id' => 'primary-widget-area',
        'description' => __( 'The primary widget area'),
        'before_widget' => '<li id="%1$s" class="widget-container %2$s">',
        'after_widget' => '</li>',
        'before_title' => '<h3 class="widget-title">',
        'after_title' => '</h3>',
        ) );
      }
  add_action( 'widgets_init', 'islam_widgets_sidebar' );

For your actual widget’s issue, you’re nesting a function inside another function, the display callback needs to be outside the function.

// Register widgetized areas
function islam_theme_widgets() {

    // Area 1
    wp_register_sidebar_widget(
        'mayeenul_widget_area',          //widget_id/ widget_slug
        'Mayeenuls Widget Area',          //widget_name
        'islam_widget_display',       //callback function
        array(
            'description' => 'The primary widget on the right side'
        )
    );

} // end theme_widgets_init

 function islam_widget_display( $args ) {

        extract($args);
        echo $before_widget;
        echo $before_title . 'My Unique Widget' . $after_title;
        echo $after_widget;
        // print some HTML for the widget to display here
        echo "Your Widget Test";

    }

add_action( 'init', 'islam_theme_widgets' );

For a second Widget you would do the same but of course with different function callback and id’s / names.

It’s really important to note that the above functions are pretty useless because the widget can only be used once in exactly 1 of the sidebars.

You really want to be extending the WP_Widget class:
http://codex.wordpress.org/Function_Reference/register_widget