How to display category name from commenter’s custom post

You can use get_the_category

This can be done by either using setup_postdata( $comment_author_post ) or by passing the post ID using $comment_author_post->ID

USING setup_postdata()

$args = array(
    'post_type' => array( 'post' ),
    'post_status' => array( 'publish' ),
    'author' => $comment->user_id,
    'orderby' => 'post_date',
    'order' => 'ASC',
    'posts_per_page' => 1
);
$comment_author_posts = get_posts( $args );
if ($comment_author_posts):
    foreach ( $comment_author_posts as $comment_author_post ):
        setup_postdata( $comment_author_post );
        $categories = get_the_category();
        ?>
        <a href="https://wordpress.stackexchange.com/questions/250201/<?php echo get_permalink( $comment_author_post->ID ) ?>"><i class="fa fa-users" aria-hidden="true"></i>
        <?php echo esc_html( $categories[0]->name ); ?></a>

        <?php
    endforeach;
endif;

USING $comment_author_post->ID

$args = array(
    'post_type' => array( 'post' ),
    'post_status' => array( 'publish' ),
    'author' => $comment->user_id,
    'orderby' => 'post_date',
    'order' => 'ASC',
    'posts_per_page' => 1
);
$comment_author_posts = get_posts( $args );
if ($comment_author_posts):
    foreach ( $comment_author_posts as $comment_author_post ):
        $categories = get_the_category( $comment_author_post->ID );
        ?>
        <a href="https://wordpress.stackexchange.com/questions/250201/<?php echo get_permalink( $comment_author_post->ID ) ?>"><i class="fa fa-users" aria-hidden="true"></i>
        <?php echo esc_html( $categories[0]->name ); ?></a>

        <?php
    endforeach;
endif;

References

get_posts()

get_the_category()