wp enqueue style on about us page

Do not use “about” as handle as it seems to be not unique and used by WordPress itself. I don’t see the list of style handles is documented somewhere. Anyway, use something unique like ‘rejaur-about’:

<?php
function load_page_styles() {
    if ( is_front_page() ) {

        // enqueue front page styles

    } elseif ( is_page('about') ) {

        wp_enqueue_style(
            'rejaur-about', // the problem was the handle
            get_template_directory_uri() . '/css/about.css',
            array(),
            1,
            'all'
        );

    }
}

add_action( 'wp_enqueue_scripts', 'load_page_styles' );

In this particular setting the styles don’t need to be registered before enqueuing them.

Off-topic: be careful with else if and elseif.