is it just an exaggeration? ...this affirmation should be present on his website
It's just an exaggeration and you are right, affirmation of that is present on his website, right in the FAQ page. Stroustrup's position on C++ complexity is clearly stated in FAQ -> Why is C++ so BIG?
Full quote of this section is presented below for your convenience, I put bold font on the part of text that directly addresses your question (Stroustrup does not think C++ is vastly complex):
C++ isn't as big as some people imagine. It's not a tiny language designed to be a minimal language for teaching, but neither are the languages people most often compare it to, such as C, Java, C#. They too are huge compared to say, Pascal as Dr. Wirth originally defined it - for good reasons, I think. The programming world is far more complex today than it was 30 years ago, and modern programming languages reflect that.
The C++ standard is 740 pages, but that includes 400 pages of library description. The language features are described (in excruciating detail) in 340 pages. Similarly, TC++PL is 1000+ pages, but only 350 of those are devoted to the explanation of language facilities and their use; the rest discuss libraries, programming techniques, etc.
C++ directly supports (i.e., in the language) what some other languages support through libraries, so the language part will be relatively larger. On the other hand, if you want to write a "typical modern application", you need to consider operating system interfaces, GUI, databases, web interfaces, etc. the sum of language features, libraries, and programming conventions and standards that you must become familiar with dwarf the programming language. Here, C++'s size can be an advantage as far as it better supports good libraries.
Finally, the days where a novice programmer can know all of a language are gone, at least for the languages in widespread industrial use. Few people know "all of C" or "all of Java" either and none of those are novices. It follows that nobody should have to apologize for the fact that novices do not know all of C++. What you must do - in any language - is to pick a subset, get working writing code, and gradually learn more of the language, its libraries, and its tools. For my suggestion on how beginners can approach C++, see Programming: Principles and Practice using C++.
The misquote your friend refers to likely originates from that very section quoted above, as a satire on the statement "the days where a novice programmer can know all of a language are gone, at least for the languages in widespread industrial use".
Note that if you just drop the word novice from that quote, you can pretend that "can't know all of the language" applies to Stroustrup himself.
Note though that the way it is really stated (with novice word in it), it suggests that Stroustrup believes to know all of a language and even more, he believes that with sufficient experience (sufficient to stop being novice), anyone can know.