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So, how does browser fingerprinting work. Does it basically look at the 1) IP and 2) Browser Agent pair for near uniqueness?

Lots of things from browser settings to what plugins you have.

These guys will tell you how unique is your browser fingerprint

This should be mentioned here, too https://panopticlick.eff.org/

fingerprintjs readme does a good job of explaining it: https://github.com/Valve/fingerprintjs2