They definitely used to, but haven’t for a long time. It’s been viewed as an unreliable and poor practice, especially with browsers like Safari and Firefox which have already disabled 3rd Party Cookies for some time now (or at least providing the option to, as a privacy feature).
Now CORS, OAUTH, and similar mechanisms do a better, more private, and more secure job of sharing state and authentication across domains and groups of services.
Except Siri processing is actually done on your device, as of iOS 15. Which kind of blew my mind when it was announced.
Nothing is sent to Apple unless you request an online service (such as weather, maps, etc.) or unless you allow your recordings to be sent.
Try it: in airplane mode on an iOS 15 device: Siri still works at a basic level. Language processing happens locally.