It doesn't. C++'s shared pointers use atomics, just like Rust's Arc does. There's no good reason (unless you have some very exotic requirements, into which I won't get into here) to implement shared pointers with mutexes. The implementation in the blog post here is just suboptimal.
(But it's true that C++ doesn't have Rust's equivalent of Rc, which means that if you just need a reference counted pointer then using std::shared_ptr is not a zero cost abstraction.)