How to hide the login / profile urls on top of comment box for a post

To hide with CSS, add this to your theme’s style.css, or the Additional CSS section of the customizer.

.logged-in-as {
    display: none;
}

Or, to remove from the template completely, add this to your theme’s, or preferably a child theme’s, functions.php file:

add_filter( 'comment_form_logged_in', '__return_empty_string' );