No, that's not really related. Cross-site scripting's name comes from the vulnerabilities which allow an attacker to insert a <script> tag pointing at a script on another domain (or an inline script). It doesn't have to do with cookies and doesn't get around or really interact with the "block 3rd party cookies" setting.

That requires exploiting an XSS vulnerability in the target domain however. Such vulnerabilities are sadly common, but can be prevented.