The proxy is by default running in App Engine under the responsibility and control of the website owner, so I'd presume it would be handled the same as any othe

The proxy is by default running in App Engine under the responsibility and control of the website owner, so I'd presume it would be handled the same as any other PaaS or IaaS service a company uses. The data sent Google products, like Analytics, via the proxy would still be subject to GDPR as it would if sent directly from the client. Note that they do give website operators the option of running the proxy in their own environment, it's made available as a Docker image.