Google wants to track your every move and using your IP Address is one way to do that. Haven't you ever used Safari's Privacy Report page to view which Trackers it has blocked from tracking you. Currently on my device, Google is the source of the top 5 trackers that were blocked by Safari and those appeared on 484 websites I visited.
You have two completely different business models between those 2 companies. Apple generates its revenue off the devices it sells and Google offers it services for free in exchange for your personal data to sell to advertisers. It is not really being caught in the middle, it is a choice you make on whether you want your Privacy & Security protected from a company or not. That would be a great question for Google for why it is necessary to track your movements and why they popup a warning when it is not possible with a hidden IP Address.
It would seem inappropriate for Apple to protect your Privacy with Private Relay for every company except Google. As for possible solutions, you can allow less restrictive IP location detection when using Private Relay to see if that helps with the Google alert you see. Go to Settings > [your name] > iCloud > Private Relay > IP Address Location, and choose Maintain General Location.
Protect your web browsing with iCloud Private Relay on iPhone - Apple Support
