Author functions don’t work in customizer’s selective refresh

The selective refresh request doesn’t get the whole regular WordPress request context. That is why on initial load (regular request) it works, while on AJAX partial refresh it doesn’t. In your case, the global $authordata is not set and get_the_author_meta() is relying on it when no user is explicitly provided.

You will need to do a little bit of work yourself:

function refresh_callback() {
    // First get the current post
    $current_post = get_post();
    // Test if we actually have a post content (we might be in an archive context)
    if ( ! empty( $current_post ) ) {
        echo get_the_author_meta('ID', $current_post->post_author );
        echo get_the_title( $current_post );
    }
}