If you use (like iCloud Private Relay or Limit IP Address Tracking), they don’t directly block cookies — but browser settings are what matter.
Safari third-party cookies by default. You cannot fully allow them in recent versions, but you can disable Prevent cross-site tracking (which may allow some session cookies): how to allow third party session cookies on mac
Google Chrome is in the process of phasing out third-party cookies entirely, but for now, you can still control them manually. If you use (like iCloud Private Relay or