There is a lawsuit claiming Google tracks users in private browsing mode. How exactly is Google achieving this?
There is a common misunderstanding of what "Private Browsing" or "Incognito Mode" actually does.
There is no anonymity or tracking protection provided. What these modes do is block persistence of information across sessions, preventing permanent storage of cookies or history after the browser has been closed.
Within a given browser session, meaning from start to stop, Incognito and similar works exactly the same (caveat) as a normal browsing session. The difference is that local storage is blocked so all of the session information is gone upon browser termination.
Some browsers (Firefox) proactively block writing to local storage in private mode. Some browsers (IE as of 2 years ago) wrote to local storage but cleared local storage upon start up.
Because protected browsing may not allow HTML5 local storage, many sites now complain about protected browsing. Some sites, like Netflix, will not work at all without the ability to write local storage.
Google is able to track users as they browse from site to site - even if the user is using Google Chrome in incognito mode - if the sites that the user visits have Google's services installed, such as Google Analytics, Google Ad Manager, etc.
This is similar to how Facebook is able to track users as the move from site to site - even if the user is using a browser in incognito mode (or private mode), and even if the user is not a Facebook user - if the sites have the Facebook 'Like Button' installed. See How does Facebook track your browsing without third party cookies? for more information.
Also, see https://threatpost.com/google-faces-privacy-lawsuit-over-tracking-users-in-incognito-mode/156269/, which explains:
The lawsuit, filed in the federal court in San Jose, California, alleges that Google compiles user data through Google Analytics, Google Ad Manager and other applications and website plug-ins, including smartphone apps, regardless of whether users click on Google-supported ads, according to a report in Reuters.