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.
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.