Create a drop down list within an active page not nav menu

I’m not sure if you have pages with categories, but I can show you a example using posts instead.

Dropdown Filter from posts by category:

<?php

    $url = get_home_url();
    $category_slug = 'my-new-category'; //remember to get the slug


    $posts_by_category = new WP_Query([
        'post_type' => 'post',
        'category_name' => $category_slug,
        'nopaging' => true
    ]);


    if ( $posts_by_category->have_posts() ) :
        $dropdown = '<select onchange="window.location.href = this.value" name="posts-by-category">';

        while ( $posts_by_category->have_posts ) : $posts_by_category->the_post();

            $dropdown .= sprintf('<option value="%s?p=%s">%s</option>', $url, get_the_ID(), get_the_title() );

        endwhile;

        $dropdown .= '</select>';

    endif; wp_reset_postdata();

    echo $dropdown;

?>