Problem with DRUPAL_UID with multiple domain-based Drupal sites

In the scenario where multiples sites under the same domain use a domain-based $cookie_domain, as opposed to site-based, every instance receives the DRUPAL_UID cookie, which is output for the cookie_domain.

However, if these sites have not configured session sharing, this means every site receives incorrect information, which can disturb caching optimizations.

It would be better if the name of the cookie could be customized per-site.