Create alphabetical Pagination for custom taxonomy?

In your code you are showing “A” for all the paginations and you also don’t need to create link for every letters in English. I fixed them. And I also added a class here called selected to current link in pagination link groups by which you can design the current link in pagination group. Please analyze the code first and write your CSS according to it. Here is the updated page code-

<?php 
get_header();
$name_like = isset($_GET['character']) ? $_GET['character'] : '';
$letters = range('A', 'Z');
?>

    <div id="content-small">
        <div class="list-terms">
            <h1 class="list-title">Listing all Tags available</h1>

            <div class="alphanav">
                <?php foreach ($letters as $letter): ?>
                    <?php if (strtoupper($name_like) == strtoupper($letter)):?>
                        <a class="selected" href="https://wordpress.stackexchange.com/questions/250166/<?php bloginfo("url'); ?>/genre/?character=<?php echo strtoupper($letter)?>"><?php echo strtoupper($letter)?></a>
                    <?php else: ?>
                        <a href="https://wordpress.stackexchange.com/questions/250166/<?php bloginfo("url'); ?>/genre/?character=<?php echo strtoupper($letter)?>"><?php echo strtoupper($letter)?></a>
                    <?php endif;?>
                <?php endforeach;?>
                <a href="https://wordpress.stackexchange.com/questions/250166/<?php bloginfo("url'); ?>/genre/?character=#">#</a>
            </div>

            <?php $args = array('name__like' => $name_like, 'order' => 'ASC');
            $terms = get_terms('category', $args);
            if ( !empty( $terms ) && !is_wp_error( $terms ) ) {
                $count = count($terms);
                $i=0;
                $term_list="<ul class="my_term-archive">";
                echo '<h2 class="term-letter">'. strtoupper($name_like) . '</h2>';
                foreach ($terms as $term) {
                    $i++;
                    $term_list .= '<li><a href="' . get_term_link( $term ) . '" title="' . sprintf(__('View all post filed under %s', 'my_localization_domain'), $term->name) . '">' . $term->name . '</a></li>';
                    if ($count != $i) {
                        $term_list .= '';
                    }
                    else {
                        $term_list .= '</ul>';
                    }
                }
                echo $term_list;
            }
            ?>
        </div>
    </div>

<?php get_footer(); ?>

And I also think you don’t need <a href="https://wordpress.stackexchange.com/questions/250166/<?php bloginfo("url'); ?>/genre/?character=#">#</a> but I kept this in case you need it.

Hope that helps.