Is it possible to edit the styling of the admin panel from within a custom theme?

Everything you’re able to do in a plugin you could do in a theme. It’s more like that separation is preferred, see article: https://managewp.com/blog/themes-design-plugins-functionality

Below an example adding editor stylesheet:

/**
 * Registers an editor stylesheet for the theme.
 */
add_editor_style( array(
    'assets/css/bootstrap.min.css',
    'https://use.fontawesome.com/releases/v5.3.1/css/all.css',
    'style.css',
));

Below an example to add scripts to the WP admin backend.

function admin_scripts() {
    wp_enqueue_style( 'bootstrap', get_theme_file_uri( '/assets/css/bootstrap.min.css' ), array(), null );
    wp_enqueue_script( 'bootstrap-bundle-js', get_theme_file_uri( '/assets/js/bootstrap.bundle.min.js' ), array('jquery'), null, true );
}
add_action( 'admin_enqueue_scripts', 'admin_scripts' );

If you need to load your scripts earlier for whatever reason, try admin_init below

add_action( 'admin_init', 'admin_scripts' );