Yes, the browsers do share parts of their JavaScript execution engine with WASM. Which makes sense because WASM still needs to interact with JS. That doesn't mean it can't be fast or start quickly (or at least quicker than asm.js).

From the developer's point of view that doesn't matter. You're delivering WASM and not C or JS code over to the clients.

WASM requires a much simpler VM than JS. It doesn't take most of a JS VM.