They're discarded on a per-session basis.

Using the 'New Tor Circuit for this Site' will only cause Tor Browser to use a new circuit for the site, it will not clear any other information. That is what the 'New Identity' option is for, see the [Tor Browser Design and Implementation](https://www.torproject.org/projects/torbrowser/design/#new-identity).

**N.B.** On Firefox's *Private Browsing* windows, cookies and other tracking data is absolutely not removed when a *tab* is closed. It is unset when the *window* is closed. This is trivially testable by visiting a site that sets and displays a cookie, closing the *tab* (but keeping the *window* open) and opening a new *tab* on the *Private Browsing* *window* and then checking to see if the cookie is still present.

When tested, it can be seen to be still set. The same is true of Chrom{e,ium}s *Inconito Window*.