Adding a custom field to the site identity menu

It seems by default WordPress provides fields for site-title and tagline.

I understand how to add (for example) the option to chose a logo, adding add_theme_support( 'custom-logo' ); to the functions.php file. However, how do I add fields for ‘company name’ and ‘company division’ or for any text field that is not already part of the WordPress theme support?

You’ll have to add your own customizer controls to achieve that.

So for example, if you want to add Company Name, you can use this code:

function my_register_additional_customizer_settings( $wp_customize ) {
    $wp_customize->add_setting(
        'my_company_name',
        array(
            'default' => '',
            'type' => 'option', // you can also use 'theme_mod'
            'capability' => 'edit_theme_options'
        ),
    );

    $wp_customize->add_control( new WP_Customize_Control(
        $wp_customize,
        'my_company_name',
        array(
            'label'      => __( 'Company name', 'textdomain' ),
            'description' => __( 'Description for your field', 'textdomain' ),
            'settings'   => 'my_company_name',
            'priority'   => 10,
            'section'    => 'title_tagline',
            'type'       => 'text',
        )
    ) );
}
add_action( 'customize_register', 'my_register_additional_customizer_settings' );

PS. Here you can find more docs regarding this topic: Theme Customization API