<form action="<?php echo site_url() ?>/wp-admin/admin-ajax.php" method="POST" id="filter">
            <?php
            if( $terms = get_terms( array(
                'taxonomy' => 'category', // to make it simple I use default categories
                'orderby' => 'name'
            ) ) ) :

                echo '<select name="categoryfilter">';
                echo '<option name="all" value="">Все категории</option>';

                foreach ( $terms as $term ) :
                    echo '<option name="'. $term->name .'" value="' . $term->term_id . '">' . $term->name . '</option>'; // ID of the category as an option value
                endforeach;
                echo '</select>';
            endif;
            ?>

            <select name="orderby" id="orderby">
                <option
                    value="date"
                    <?php echo selected($_GET['orderby'] ?? '', 'date'); ?>
                >
                    По новизне
                </option>
                <option
                    value="down"
                    <?php echo selected($_GET['orderby'] ?? '', 'down'); ?>
                >
                    По убыванию
                </option>
                <option
                    value="up"
                    <?php echo selected($_GET['orderby'] ?? '', 'up'); ?>
                >
                    По возрастанию
                </option>
            </select>

            <input
                    id="order"
                    type="hidden"
                    name="order"
                    value="<?php echo (isset($_GET['order']) && $_GET['order'] == 'ASC') ? 'ASC' : 'DESC'; ?>"
            >

            <input type="submit" value="Применить">
        </form>