You send your AJAX request to /wp-admin/admin-ajax.php (which is very good), so it’s not a typical WP request.
During such request, request is not parsed, no global WP_Query is created, and so on.
is_author is a Conditional Tag, which checks if an Author archive page is being displayed. So, by design, it will return false during AJAX request, because AJAX request is not displaying author archive page…
On the other hand, is_user_logged_in() works, because user is still logged in during AJAX request (it’s the same session).
PS. It’s the same with get_query_var – no query is parsed or executed, so there are no query variables available…