That stuff is still around unless you bleach bit it or melt the platters.
Some truth to that. If you've ever tried to get around the WSJ or NYT paywall, you'll know what just an outsourced cloud provider can do for the dying dead-tree media. Clearing cache and cookies isn't enough. Even visiting it from a different browser on the same PC you've never used before is good enough. Never mind what the government or a really determined party in Silicon Valley could do.
There's a lot of identifying information that can be had, assuming any of the browser apps have an API in them that can reach those parts of the machine. Even if you're running from a write-protected USB boot disk with a Linux distro on it. And you use a different public IP and you spoof your MAC address.
For several years now, all Intel processors, probably others, and probably certain other things on the PC's bus and chipset have unique identifying serial numbers in them that can be queried by various applications poking certain memory registers. Mainly it was intended for trying to get some low level DRM schemes on DVD's, Blu-Ray disks, video codecs etc. But it's really useful for tracking a PC and getting it's "fingerprint" too.
