CAPTCHA's are a little tricky to explain, and no two CAPTCHA's or DDoS protection services are the same, so I'll try to give a more general overview of it.
Every CAPTCHA has a unique ID number attached to it, and when you solve the CAPTCHA, your IP and all of your browser details are, in theory, shipped back to the CAPTCHA server attached to this unique ID. I don't know how Google does this, but Cloudflare does this with RayIDs.
What I'd really like to drive home here is that you can be identified by a whole lot more than just your IP address. I'd like for you, and any other interested anons, to check some of these links out to see exactly what can be tracked in your browsers.
Your IP address is one thing: https://browserleaks.com/ip
can track a lot more details, like your screen size, time zone, operating system, battery life, even what extensions you have installed.
You can track what your graphics card can do: https://browserleaks.com/webgl
You can track your system based on how you display images: https://browserleaks.com/canvas
You can even track if someone is using Tor or an ad blocker! https://browserleaks.com/proxy
Oh, and private mode does absolutely nothing: https://www.nothingprivate.ml/
Want to be super spooked? Look no further than this: https://clickclickclick.click/
Anyways, the point here is that even if you did
change your IP address, through either Tor or a VPN, there's a dozen other ways to track you that can be combined and 'triangulated' to break your anonymity. A CAPTCHA is a very easy way of doing this, since they have to be able to phone home to do their jobs, and it would be very easy to embed tracking scripts into it. With all that data attached to that ID number, it would be a cinch to trace you. Tor Browser can only do so much.
If you think ad companies drool over that kind of data, imagine what the Feds must be thinking.
Even if you block the CAPTCHA, well, you just can't access the site. Do that on the Tor browser, and you create a unique browser signature that means you can be tracked more easily everywhere else. That's why they don't include an ad blocker in the Tor browser, by the way.
Pretty crazy, huh?