How do i add a unique body class to the wordpress dashboard’s home page?

If I’m not mistaken, you could just use body.wp-admin.index-php to target the dashboard.

If you want to add something else, you can use the admin_body_class filter to manipulate the classes added to the body and check what is displayed with get_current_screen:

add_filter("admin_body_class", function($classes) {
    $current_screen = get_current_screen();
    if($current_screen->base == "dashboard") {
        $classes .= " dashboard";
    }
    return $classes;
});