> But is there anything that prevents third-party analytics and similar services from creating libraries or services that are deployed under their clients' domain, and then transmit the collected data from client site to third-party service's own database?

No, because to do this unambiguously is simply impossible. It is fundamentally always technically possible for your own site to track user behavior of people on your site (what's to say what's "tracking" versus simply serving site page data), and there is no technical way to prevent a site owner from simply sharing their own tracking data with data aggregators. There are of course some heuristics that tracking blockers could use (e.g. "Even though this script is served from somesite.com, it looks like the Facebook tracking code"), but that just becomes a cat-and-mouse game to obfuscate the script.

That's a big reason I feel this "third party cookie blocking" was just oversold. Adding third party cookies was just technically easy because it just involved adding a simple JS snippet to a page, but serving that snippet from one's own domain is still pretty trivial, and indeed what most of the big ad brokers are moving toward.