thats a good question indeed.
I'm a tea beginner myself, so I cannot definetly answer your question, but I'll try anyways, by adding some thoughs on the subject which may be more or less correct / helpful.
First I'd say that all these categories do their job, to categorize teas, very well for most teas. But these categories combine different aspects like cultivar, production technique, appearance, aging and more, and as always there are some candidates which are not as easily classified, so I wouldn't expect these classifications to be the last word on the subject.
Still, globally, I only heard the term "Sheng" in reference to Pu'er, to be more exact, to distinguish "pile-fermented" Pu'Er - Shu - from the old school stuff
And, for the sake of completeness, while "white tea" is such a category, for Sheng the category would be "hei cha - dark tea", from which Pu'Er is a subitem, being the most prominent one of course.
Another thought is that you can of course press virtually anything to cakes, so that may not account for much. There are red (= in western terms, black) tea cakes, oolong cakes, so why not white tea cakes. I'm still waiting for a nice Bing of Gyukuro though
Also, maybe White tea is not so extremely much different from Pu'er, because they're both rather simply processed teas compared to, say, Oolong. Withering, kill-green, drying, that's more or less it. Of course, there is a world of variations possible here, hence this is a very, very coarse statement, if that comparison is possible at all!
Still, I find it rather surprising that anyone would mix these two labels up in the first place. Would be interesting to know where you read that, its well possible that this person knew very well what he (she) was saying.
And for the enzymes: If I'm correct, Pu'Er does also undergo the "sha quing" process to kill off enzymes, but that technique is carried out in a more careful manner than with green tea in order to let SOME of the enzymes survive the process. I believe that if that isn't done at all, the tea will oxidize and become very similar to "red tea" (Hong Cha, which is called black tea in the west).
Heres a nice and short article on the subject: Tea Urchin - How is Pu'Er processed differently from other teas
Finally, I say again that I'm an amateur on the subject and that I'm not qualified to seriously answer your question. I mainly spit out what I've read along the way from sources I believe to be trustworthy. If anyone who has more first-hand knowledge on the subject would contribute and / or correct, it would be great.