Wordpress - Including categories in search results

Use get_terms(), then you don't need to use a custom query to the database.

$terms = get_terms( 'category', array(
    'name__like' => $s,
    'hide_empty' => true // Optional 
) );
if ( count($terms) > 0 ){
    echo '<ul>';
    foreach ( $terms as $term ) {
        echo '<li><a href="' . esc_url( get_term_link( $term ) ) . '" title="' . esc_attr( $term->name ) . '">' . esc_html( $term->name ) . '</a></li>';
    }
    echo '</ul>';
}

Based on birgire's answer on a similar question: https://wordpress.stackexchange.com/a/239680/50432

I'm using this code in my search.php above the main loop:

$search_term = explode( ' ', get_search_query( false ) );   
global $wpdb;
$select = "
SELECT DISTINCT t.*, tt.* 
FROM wp_terms AS t 
INNER JOIN wp_term_taxonomy AS tt 
ON t.term_id = tt.term_id 
WHERE tt.taxonomy IN ('category')";      
$first = true;
foreach ( $search_term as $s ){
    if ( $first ){
        $select .= " AND (t.name LIKE '%s')";
        $string_replace[] = '%'.$wpdb->esc_like( $s ).'%';
        $first = false;
    }else{
        $select .= " OR (t.name LIKE '%s')";
        $string_replace[] = '%'. $wpdb->esc_like( $s ).'%';
    }
}
$select .= " ORDER BY t.name ASC";
$terms = $wpdb->get_results( $wpdb->prepare( $select, $string_replace ) );
if ( count($terms) > 0 ){
    echo '<ul>';
    foreach ( $terms as $term ) {
        echo '<li><a href="'.esc_url( get_term_link( $term ) ).'" title="'.esc_attr( $term->name ).'">' . esc_html( $term->name ) . '</a></li>';
    }
    echo '</ul>';
}

This code does an extra DB query, but search for categories not only associated to returned posts, but does an extra seach for each word in a search term and brings all found categories - even empty.