> but they’re not saving that, right?If you are logged into your Google account in the "private" session, your private session will carry session-level data (i.e Google's cookies) that will persist until the last private window is closed and that data can and will be associated with your account.
So if you are logged in to Google, even in a private window, they will be tracking you through and and all of their properties. If you don't want to be tracked as you in private windows, don't log in as you in private windows.
This part isn't Google (or anyone else in their position)'s doing, it is by browser design. Browsers do what they can to make it difficult to detect the difference between normal and "private" windows so Google can reliably detect is that at the start of the private session you have no Google cookies, which is the same behaviour if you had moved to using a fresh new browser (in private mode or not). It is currently possible to work out that you are in private mode in many browsers using various tricks (some browsers/versions disable cookies in private mode in a manner detectable via navigator.cookieEnabled, some disable all local storage options in a way that can be detected by exceptions firing when you try use them, some disable service workers, ...) but none of these tricks are 100% reliable (even, I suspect, if you try use them all) and may all become 100% unreliable at a later date. If they can't reliably detect you are in a private window, then they can't reliably turn off tracking of your logged-in activity in private windows.
> They can definitely tell it’s the same user who just opened a new private tab
This is a slightly different matter. Here the link may contain information that can be used to pollute the private island. So they know opening that advert link in a private window was done by you because the URL contains a code that was only given out to use (in a non-private window). But again, if they can't reliably detect the privacy settings they can't be expected to do anything about them.
> Google associates your browsing history with your Google account when your not logged in and using private browsing.
This could partly be the case, though not in a practically avoidable manner. This is the problem because people will assume that Google and their ilk can't continue to track you, not that it is difficult to know they should stop tracking you, when you do something like right-click-open-in-private-tab. The trick is to never log into your account in a private window and never follow links from non-private windows in private ones (without verifying they don't carry personal/session data first), but try explaining that to the average user who barely knows what a cookie is.