As a fundamental aspect of Privacy Browser's design, web pages are not allowed to open other tabs or otherwise modify the UI of anything outside of their own window. You can read a bit about this at:
https://www.stoutner.com/privacy-browser-android/core-privacy-principles/
Any request to open a new tab is simply opened in the current tab.
That shouldn't really cause a problem in this case, so the root of the issue is something else. The most likely problem is probably that the website will not allow downloads of PDF files if it doesn't receive a referrer header. Privacy Browser never sends a referrer header for a main frame URL, as a website never has any right to know which web page you visited before requesting the current URL.
There is some information about the referer header at:
https://www.stoutner.com/the-referer-header/
There is also a feature request regarding blocking of the referer header on resource requests and creating a domain setting to control it at Feature #586: Disable the Referer header on resource requests by default (although I find the referer header so repugnant that I might just disable it completely with no method of enabling it).
In any regard, I would consider the download behavior on this website buggy. If I were you, I would contact the webmasters and ask them to fix their buggy behavior so PDF downloads worked like their EPUB downloads.