delete_term is not working properly with add_action()

AFAIK, slug is unique. So $post_cat_term = get_term_by( 'slug', $deleted_term->slug, 'category' ); shouldn’t return anything.

Instead, you may try name.

Also, delete_term callback gets 5 parameters, so it’s better to adjust that accordingly.

See if the following works:

function delete_similar_term( $term_id, $tt_id, $taxonomy, $deleted_term, $object_ids ) {
    if( $deleted_term->taxonomy === 'movies_category' ) {
        $post_cat_term = get_term_by( 'name', $deleted_term->name, 'category' );
        wp_delete_term( $post_cat_term->term_id, 'category' );
    }
}
add_action( 'delete_term', 'delete_similar_term' , 10, 5 );