This of course doesn’t help explain why Firefox is so heavily targeted by what’s supposed to be a neutral utility like Google Analytics...
The idea of tracking your history across multiple reCAPTCHA loads across multiple domains to build a user profile is what sounds like a giant privacy red flag, even though it's entirely possible given the current implementation.
Additionally asking hosts to include JS directly onto their domain which sets 3rd party cookies/data across every page in addition to tracking referring domains is equally a bad idea. reCAPTCHA 2/3 does require loading 3rd party JS directly on page, which I'd imagine is necessary to create callbacks in the frontend upon verification (as iframe content messaging is very awkward):
Ideally the JS simply loads an iframe of the captcha HTML and handles the callbacks from events in the iframe. That's it. It shouldn't be touching anything else on your website. I'd be curious to see a reverse engineering to see how much the JS really does...
Yeah, no. It certainly can read non-google cookies on the page (not httpOnly cookies, though).
That said, I've no evidence one way or the other!
My understanding is that it comes down to information they can read about your browser (does this look like a bot environment?), and heuristically how the user has behaved since the JS has been loaded (mouse movements, time between actions, etc).