Private Access Tokens are built into Safari. Apple gives their devices a certain amount of tokens, and Cloudflare validates them. If Apple doesn't like your device, you won't get any more tokens. Cloudflare also hands out tokens if you install their browser addon.
The two companies are working together to make this an official web standard, but I haven't heard about it for a while. Maybe they're just laying low after seeing the blowback on Google's (worse) attempts at attesting devices.
Is it really built into Safari, or are you referring to the iCloud option that "privately" helps reduce Cloudflare demands?