Actually all three of Rust's user defined types (the sum type enum, the product type struct, and union†) are fully fledged types which can implement traits and have functions of their own (including functions taking a self parameter, thus methods)

C++ unions can actually have methods, although this isn't used very much. However C++ enums can't have methods, even C++ 11 scoped enums ("enum classes") can't have methods, I have no idea why that restriction seemed like a good idea.

† Unions are special because they're crazy dangerous, which is why they're not usually covered in material for learning Rust - you can't fetch from them safely. You can store things in unions safely because the process of storing a value in a union tells the compiler which is the valid representation - the one you're storing to, but fetching is unsafe because you might fetch an inactive representation and that's UB. However Rust does have a particularly obvious union right in the standard library - MaybeUninit - and sure enough MaybeUninit implements Copy and has a bunch of methods.