Modifying the way WP sets cookies

Short answer: You can’t. That’s not hookable for the settings cookies.

You can adjust the cookie paths and such via defines, so as to make the settings apply across multiple installations across a domain. The most common use of this is to make logins work across a domain, so when you log into one, you log into all subdomain installs as well.

The basic code you need here is this in the wp-config file:

define('LOGGED_IN_COOKIE', 'login_cookie_name');
define('AUTH_COOKIE','auth_cookie_name');
define('COOKIE_DOMAIN', '.example.com');
define('COOKIEHASH', 'random_hash_here');

Put that in the config across multiple sites, set the keys and salts to be the same, and you’ll have login cookies that work across the domain and subdomains.

Multisite is simpler.