There are only two cookie jars in Firefox, one for regular mode and one for private browsing mode tabs. In regular mode you have multi-account containers that use their own cookies if you need to isolate session data. On Mac you would have to quit the Firefox application instead of closing all open windows if you use permanent Private Browsing mode (Always use Private Browsing mode; Never Remember History) to clear the PB mode cookie jar.

Firefox Multi-Account Containers: https://addons.mozilla.org/firefox/addon/multi-account-containers/

Yep. There is no such thing as "private" browsing.

When I intentionally open a new private browsing window, this new window detects what websites I have open in other "private" windows. For instance, say I'm logged into my favorite newspaper and I find an interesting link, say, on youtube.

I should be able to open a new private browsing window and paste that link into the address bar without Google (youtube) knowing all the details of my logged-in status in the other private browsing window. This is not the case. Tabs AND windows share data and there is no way to isolate and keep private information in other tabs or windows.

I always use private browsing I do not allow third-party cookies I always block trackers I always use a VPN

GOOGLE knows who I am by looking at log-in status in other windows or tabs

If you want to separate tabs from using the same cookie jar then you would have to run these tabs in different containers instead of using Private Browsing mode. PB mode only means no data is stored to the hard drive and instead is kept in memory and nothing more. I don't think or are sure that you can combine PB mode and containers, so you will have to choose foe either.