Adds the ability to display the name of the related term in the Custom Field area of Ajax Search. In the Key field, enter post_terms__category where the 'category' part should be replaced with the taxonomy slug.

display_terms_in_ajax_search_meta.php

<?php

add_filter( 'jet-search/template/pre-get-meta-field', function($value, $meta, $post){
    $meta_key = $meta['meta_key'] ?? '';

    if( false === strpos( $meta_key, 'post_terms__' ) ){
        return $value;
    }

    $tax_slug = ( explode('post_terms__', $meta_key) )[1];

    if( ! taxonomy_exists( $tax_slug ) ){
        return $value;
    }

    $terms = wp_get_post_terms( $post->ID, $tax_slug, array('fields' => 'names') );

    return array( implode(', ', $terms) );

}, 10, 3);