how to indicate on post listing title if user already liked this post

Judging by the code for that like system, you should be able to find out if the user has liked the post like this:

if ( is_user_logged_in() ) {
    $likes = get_user_meta( get_current_user_id(), '_liked_posts', true );

    if ( is_array( $likes ) && in_array( get_the_ID(), $likes ) ) {
        echo '<span class="liked">Liked</span>';
    }
}

That will echo a <span> with the class liked if the user has liked the post. That code will need to go in the template where you want the “Liked” badge to appear.