Display post category in foreach loop

When the foreach loop is from get_posts(), you need to run setup_postdata() for each post if you want to use themplate tags inside the loop like the_category(), if not you can get unexpected behaviour. Also, you will need to run wp_reset_postdata() after the foreach loop. Note that the_category and similar template tags actually displays/echoes the output, so, if you want to concatenate the output in a string you should use get_the_category() instead.

function my_get_display_author_posts() {
    global $authordata, $post;

    $authors_posts = get_posts( array( 'author' => $authordata->ID,'posts_per_page' => 6, 'post__not_in' => array( $post->ID ) ) );

    $output="<ul>";

    foreach ( $authors_posts as $authors_post ) {

        // Build a comma separated categories list
        // You can customize as needed
        // or use get_the_category_list() for quick and delimited list of categories
        // http://codex.wordpress.org/Function_Reference/get_the_category_list
        $categories = get_the_category($authors_post->ID);
        $categories_string = '';
        $separator=", ";
        if($categories) {
            foreach($categories as $category){
                $categories_string .= '<a href="'.get_category_link( $category->term_id ).'" title="' . esc_attr( sprintf( __( "View all posts in %s" ), $category->name ) ) . '">'.$category->cat_name.'</a>'.$separator;
            }
            $categories_string = trim($categories_string, $separator);
        }

        $image = wp_get_attachment_image_src( get_post_thumbnail_id($authors_post->ID), 'related-author' );
        $output .= '<li>
        <a class="title" href="' . get_permalink($authors_post->ID) . '">
        <strong>' . apply_filters( 'the_title', $authors_post->post_title, $authors_post->ID ) . '</strong>
        <img src="'.$image[0].'"> 
        </a>
        <span>'.get_the_time('m.d.y', $authors_post->ID ).'</span>'.$categories_string.'
        </li>';
    }

    $output .= '</ul>';

    return $output;
}