functions.php

<?php

add_action('wp_ajax_addTagToStory', 'add_tag_to_story');
add_action('wp_ajax_nopriv_addTagToStory', 'add_tag_to_story');

function add_tag_to_story() {
    // Here you have everything passed from your JS function available as $_POST
    // With $_POST['postID'], you can do almost any post manipulation you need
    var_dump($_POST);
    
    
    if(validate_nonce($_POST['nonce'], 'iwt_add_tag_to_post_' . $_POST['postID']) {
        wp_set_post_terms( $_POST['postID'], 'my_new_tag_slug', 'tag', false );
        
        echo "<div class='alert alert-success'>Successfully added tag to post!</div>";
        
        die(); // important! 
    } else {
        // nonce failed, send error message
        echo "<div class='alert alert-danger'>Nonce validation failed, please try again</div>";
        
        die();
    }
    
}

function create_nonce($nonce_string = 'iwt-nonce') {
    return wp_create_nonce($nonce_string);
}

function validate_nonce($nonce, $nonce_string = 'iwt-nonce') {
    if (!wp_verify_nonce($nonce, $nonce_string)) {
        // This nonce is not valid.
        return false; 
    } else {
        // The nonce was valid.
        return true;
    }
}



?>

loop.php

<?php
$args = [
    'post_type' => 'story',
    'posts_per_page' => 10,
    'orderby' => 'title',
    'order' => 'ASC'
];
    
$query = new WP_Query($args); ?>

<?php if($query->have_posts()) : while($query->have_posts()) : $query->the_posts(); ?> 
    <div id="post-<?= get_the_ID() ?>" class="append-our-message-here">
        <h1><?= get_the_title() ?></h1>

        <a href="javascript:;" class="click-me-to-add-tag" data-id=<?= get_the_ID() ?> data-nonce=<?= create_nonce('iwt_add_tag_to_post_' . get_the_ID()) ?>>Click me to add a tag with AJAX</a>
    </div>
<?php endwhile; endif; wp_reset_postdata(); ?>

scripts.js

(function($) {
    $(document).ready(function() {
        addTagToStory();
    });
    
    
    function addTagToStory() {
        var actionButton = $('.click-me-to-add-tag');

        $(actionButton).on('click', function(e) {
           e.preventDefault();

           console.log("Running thingy");

           var postID = $(this).data('id');
           var nonce  = $(this).data('nonce);

            $.ajax({
                type: 'POST',
                url: '/wp-admin/admin-ajax.php',
                data: {
                    dataType: 'json',
                    'action': 'add_tag_to_story',
                    'postID': postID,
                    'nonce': nonce
                },
                beforeSend: function () {
                },
                complete: function () {
                },
                success: function (data) {
                    // Do your front-end stuffs here, like adding a "Success" message or a spiny wheel or whatever
                    // data is whatever was returned by the php function
                    console.log(data);
                    // This should append our alert to the DIV
                    $('#post-' + $_POST['postID']).fadeOut().append(data).fadeIn();
                },
                error: function (xhr, ajaxOptions, thrownError) {
                    alert(xhr.status);
                    alert(thrownError);
                }
            });
        });
    }
}