@afavour: The topic isn't as simple as having a HTTP header with a unique identifier. Browser Fingerprinting is a complex process, that uses unintentional implementation details, like how things are rendered with different graphics drivers or details you can get from APIs that are intended for other purposes (like WebRTC).
The site that morjom posted gives you a simple overview and Firefox is known for the privacy preserving features it comes with. However, you are right, that it is an inexact science as long as we don't know the logic of the Cloudflare implementation.
