They're rather different: In Rust types only exist at compile time; dyn Any is a normal trait object, so you can only call the trait's methods. With C#'s dynamic, you can call arbitrary methods and access any fields with type checking of those accesses being delayed until runtime, which works because types exist at runtime too.

Rust's dyn Any corresponds better to C#'s Object; dynamic exists to interface with dynamic languages and is rarely used.