Sounds like some trolling to me, especially since he can't produce the email. If Google was actually doing this they would have emailed other NewPipe users, not just this one guy.
The assurance that they've used no other apps I put little faith in; I've too often both delivered and received the incorrect "but I'm sure I did nothing else, it had to be this!" claim. We do a lot of things without forming strong memories of them all the time. No offense intended to the original reporter, it's just my experience says that's rarely a highly trustworthy claim, even when trustworthy people make it.
(Obvious solution: use the approved clients.)
One would have to do a deep-read of the implementation of those APIs to know what they might leak (or toss a packet sniffer in the network stack).
Using the system browser or webview doesn't send any user identifying information (other than device type and version) in http requests. There's no plausible reason why using the download manager API would do it either.
There isn't evidence other than the user in question's account apparently got banned and they claim to be using NewPipe. So it's one possible avenue of further investigation if one tries to figure out how Google would even know to ban a user that was using a tool that allows connection to YouTube anonymously.
But if that guy built a custom build of NewPipe that did a million simultaneous downloads, and that impacted service availability for other users, that would make Google ban him.
https://news.ycombinator.com/item?id=21248111
(I assume that's the one you meant. "Ctrl-F" is very vague when you have a bunch of comments in this thread.)