That's not intrinsically or pervasively true, although it's not uncommon.
Yes. The main areas where Rust code uses dynamic linking by default are Glibc and OpenSSL (through the popular `native-tls` crate). Most things outside of that will be statically linked by default. There is room to improve the situation by making more C wrapper libraries support either method.
For Rust code talking to Rust libraries (such as `std`), it's a totally different challenge, similar to what you face in C++. You can compile your pure Rust app in a way to divide it up into DLLs. The problem is that the polymorphism in Rust means these DLLs must all be built together. Calling a polymorphic function means code has to be generated for it. The only way around this is for your Rust library to speak the C ABI, which bloats code as building your C-style API surface will resolve all the polymorphized functions/structs, but at least gets you swappable dynamic linking.
The only way to avoid it is to be on linux with no_std, use musl statically or be on embedded. You cannot (or at least are supposed to not be able to) link to glibc statically and on every other OS you can only call syscalls via the system libraries. (Well, technically you can on most systems, it's just not stable across updates. OpenBSD will actively block it though)
Rust binaries also dynamically link to and rely on this runtime.