> The GDPR extends far beyond the US notion of PII.

That's a good thing. The US notion of PII is ridiculously naive.

It includes IP address... the fundamental glue that makes routing to and from said servers possible. Good luck being able to resolve web requests without knowing where to send the response.

IP address is both personally-identifying information and also technically required to provide computational service.

Just like your name is personally-identifying information and (usually) required to provide medical service.

But being required for service doesn't automatically mean that it can be shared with third parties. You can't share names with third parties. Why would you share IP addresses?

A name is not a requirement to render medical service, so I don't see how that example is relevant. A practioner is capable of treating patients without knowing their name. Laws may compel them to keep track of that data, but it's not strictly necessary.

And the act of connecting to a server hosted in another jurisdiction (e.g. America) would require sharing your IP. This could be directly (the entire web service hosted in the USA), or indirectly (some of the web service's assets are hosted in the USA).

If you put a CDN in-front of your web service, then that CDN will most likely be sharing your IP with the host server too. Especially if the web service wants to do something non-cacheable that they can't offer from behind the CDN.

There are many (!) types of medical treatments. Some require multiple visits. A medical practitioner needs some way to ensure that progress is maintained across multiple visits.

The internet has multiple visits too. They're just called packets instead.