Comment I think the title is a vast overstatement... (Score 2, Informative) 84
IP address is one of a large number of bits of data used to fingerprint a device and user. Changing your IP address and even browser does very little to impact how you are uniquely fingerprinted. These include (for web browsering); User Agent, HTTP_ACCEPT Headers, Browser, Plugin Details, Time Zone Offset, Time Zone, Screen Size and Color Depth
System Fonts, Are Cookies Enabled,Limited supercookie test, Hash of canvas fingerprint, Hash of WebGL fingerprint, WebGL Vendor & Renderer, DNT Header Enabled?,Language, Platform, Touch Support, Ad Blocker Used, AudioContext fingerprint, CPU Class, Hardware Concurrency, Device Memory (GB).
Now i vaguely remember a plan by Apple to spoof this information to truely prevent fingerprinting but that is separate from the private relay concept. With large amounts of data points which can be queried from your browser fingerprinting is simply about probability that any one combination of the large number of data points is unique.... which it is likely to be.