Rusts wasm32-wasi and wasm32-unknown-unknown targets don't depend on Emscripten and they've been the preferred ways of targeting wasm for years now.
There is a wasm32-unknown-emscripten target but I think that's basically on life support if not outright deprecated. It was only really a stop-gap until native tooling like wasm_bindgen got fleshed out.
wasm32-unknown-emscripten is useful if you want to build a Rust program using SDL2 and GL for the Web, for example. But yes, as you said, it's much less used. Most Rust code on the Web is not a port of existing applications (which is what Emscripten focuses on), unlike C++.
There is a wasm32-unknown-emscripten target but I think that's basically on life support if not outright deprecated. It was only really a stop-gap until native tooling like wasm_bindgen got fleshed out.