Tracking today is an interaction between cookies and pages, not really because cookies were designed to be shared between domains. Because of that, ads on web pages are a reason that information gets shared across sites. Any ad or other iFramed content that’s served on a site can get the domain name of where it’s be served from and then access the iFrame domain’s cookies, which enables tracking.
The cookie jar language might be a tiny bit oversimplifying/confusing in the sense that cookies are already separated from each other according to the cookie’s domain - they’re already in separate jars in a way, and this feature is changing how they partition the jars. Assuming even the most strict privacy settings, tracking cookies and scripts are often in iFrames and may not have the ability to directly read other cookies from the web site in your URL bar, and the web site might not be able read the tracking cookies. It’s not that cookies themselves are being shared per se. It’s that (say) Facebook is allowed to put an iFramed tracker on some site, and Facebook can then get a tracking blip when you visit that site and add it to a Facebook-only cookie. Total Cookie Protection is going to put cookies that only Facebook can see in a different jar for each separate site you visit, making it so that Facebook can’t read it’s own cookies across different sites.