Google learns about you through cookies and other tracking mechanisms, not the source address you reach google from...Yes, I'm aware of that explanation. Just that I'm not sure is it really for DDOS prevention or is it because VPN hurts google's tracking & advertisement.
My current thinking is as follows: Google search seems free, but what google need in return is to know more info about "you". Then google can make money out of that info. Using Google without leaking any info to google is like treating google like a real free service, which is not.
If you have lots of users coming from the same source address and these users don't have existing tracking cookies, google has no way to tell if you're real users or an automated script. Google makes even less from automated scripts than they do from untracked users, as they can't even show random ads to the automated scripts.
If there's one user per address it's unlikely to be automated scripts (and even if it was, the extra load isn't worth bothering about) so google just assumes that it's not.
Once the users have tracking cookies it doesn't matter that they originate from the same address, the cookies are trackable just the same as any other. The captcha is to ensure that it's a real user and not a bot. Google couldn't care less if you use a VPN or TOR, so long as you're a real user and you let them track you.
A VPN doesn't give you any anonymity from google unless you fully clear your browser cookies, caches and history every time you use it. Aside from cookies, there are many other methods of browser fingerprinting available too.
Other services also have captchas for the same reason, cloudflare for instance.