Fix the duplicate checking logic:

Your duplicate checking logic is wrong. You needed to keep the terms in an array and then check if the term already exist in the array to determine if it’s a duplicate.

So instead of if ( $dupe == 0 ) : condition and wrong loop nesting, it’ll be like:

if ( have_posts() ) : 
    echo '<ul class="filter">';
        echo '<li class="current-menu-item"><a href="#" data-filter="*">' . __( 'All' ) . '</a></li>';     
        $unique_terms = array(); // instead of $dupe = 0;
        while ( have_posts() ) : the_post();
            $term_list = wp_get_post_terms( $post->ID, 'media_blog_category', array( 'fields' => 'all' ) );
            foreach( $term_list as $term ) :
                if( ! in_array( $term->term_id, $unique_terms ) ):
                    array_push( $unique_terms, $term->term_id );
                    echo '<li><a href="#" data-filter=".media_blog_category-' . $term->slug . '">' . $term->name . '</a></li>';
                endif;
            endforeach; //term_list
        endwhile;
    echo '</ul>';
endif; //WP_Query

This should work as you’ve wanted.

A better solution:

Your CODE will work after fixing with the above logic. However, you can make it more efficient with the following CODE:

if ( is_post_type_archive( 'media_blog' ) ) :
    $args = array(
        'post_type'      => 'media_blog',
        'posts_per_page' => 16, // same as the pre_get_posts
        'post_status'    => 'publish',
        'paged'          => get_query_var( 'paged' ),
        'fields'         => 'ids' // for this specific case, all you need are the IDs
    );
    $wp_query = new WP_Query( $args );
    if ( have_posts() ) :
        echo '<ul class="filter">';
            echo '<li class="current-menu-item"><a href="#" data-filter="*">' . __( 'All' ) . '</a></li>';
                $term_list = wp_get_object_terms( $wp_query->get_posts(), 'media_blog_category' );
                foreach( $term_list as $term ) :
                    echo '<li><a href="#" data-filter=".media_blog_category-' . $term->slug . '">' . $term->name . '</a></li>';
                endforeach; //term_list
        echo '</ul>';
    endif; //WP_Query
    wp_reset_query();
endif; //media_blog