How to read individual user’s directory and display content to that specific user?

First an observation, using $upload_dir.'/USER_PHOTOS'.$user->user_login may lead to a mess of folders in wp-content/.

I suggest wp-content/user-photos/user-login/. And the following example considers it this way.

In your case, you already have the menu item, for completeness this is the submenu used for development:

add_action( 'admin_menu', 'wpse_75873_folder_menu' );

function wpse_75873_folder_menu() 
{
    add_media_page( 
        'User Photos Media', 
        'User Photos', 
        'edit_posts', 
        'so-13416177', 
        'wpse_75873_display_page' 
    );
}

And this the page displayed for the user with the contents of his folder, and a couple of auxiliary functions:

function wpse_75873_display_page() 
{
    echo '<div id="icon-upload" class="icon32"></div><h2>User Photos</h2>';

    $baseDir = WP_CONTENT_DIR . '/user-photos/';
    $baseUrl = WP_CONTENT_URL . '/user-photos/';

    // build user folder
    global $current_user;
    get_currentuserinfo();
    $userDir = $baseDir . $current_user->data->user_login . "https://wordpress.stackexchange.com/";
    $userUrl = $baseUrl . $current_user->data->user_login . "https://wordpress.stackexchange.com/";

    if( !is_dir( $userDir ) )
    {
        echo '<div style="margin-top:30px;font-size:2em;font-weight:bold;color:#f00">
            Folder does not exist!
            </div>';
        return;
    }

    // build array with files in user folder
    $files = array();

    if ( $dir = opendir( $userDir ) ) 
    {
        while ( $file = readdir( $dir ) ) 
        {
            if ( $file != "." && $file != '..' ) 
            {
                if ( !is_dir( $userDir . $file ) ) 
                {
                    // Hide files that start with a dot
                    if( !so_834303_starts_with( $file, '.' ) ) 
                    {
                        $size = so_13416177_file_size( 
                            filesize( $userDir . $file ) 
                            );
                        $files[] = array( $file, $size );
                    }
                }
            }
        }       
        closedir($dir);     
    } 

    if ( empty( $files ) ) 
    {
        echo '<div style="margin-top:30px;font-size:2em;font-weight:bold;color:#00f">
            No photos!
            </div>';
    }
    else
    {
        ?>          
        <table class="widefat">
            <thead>
                <tr>
                    <th style="width:20%">Thumbnail</th>
                    <th>File</th>
                    <th>Size</th>
                </tr>
            </thead>
            <tfoot>
                <tr>
                    <th>Thumbnail</th>
                    <th>File</th>
                    <th>Size</th>
                </tr>
            </tfoot>
            <tbody>
        <?php

        foreach ($files as $file) 
        {
            ?>
               <tr>
                  <td><!-- THIS IS NOT A THUMBNAIL -->
                    <?php echo '<img src="' . $userUrl . $file[0] . '" width="50">'; ?>
                  </td>
                  <td>
                    <a href="<?php echo $userUrl . $file[0]; ?>">
                    <?php echo $file[0]; ?>
                    </a>
                 </td>
                 <td><b><?php echo $file[1]; ?></b></td>
               </tr>
            <?php
        }

        ?>
            </tbody>
        </table>
        <?php
    } // end else
}

// https://stackoverflow.com/q/834303
function so_834303_starts_with( $haystack, $needle )
{
    return !strncmp( $haystack, $needle, strlen( $needle ) );
}

// http://www.php.net/manual/en/function.filesize.php#110739
function so_13416177_file_size( $o, $depth=0 ) 
{
    if( $o > 1024 ) 
        return so_13416177_file_size( $o/1024, $depth+1 );

    $unit = array( 'B', 'KB', 'MB', 'GB', 'TB', 'PB', 'EB', 'PB', 'ZB', 'YB' );
    return sprintf( '%.01f %s', $o, $unit[$depth] );
}

The result:

_{Based on a recent Answer in StackOverflow. The code is adapted and improved.}