There are many ways to run a WebAssembly module, besides Web, Node.js can load wasm module with --experimental-wasm-modules, standalone runtime like wasmtime and lucet can run it too.
So how do I detect the current environment in WASM and is it possible to apply a restriction on wasm modules that make it can only work on a specific website?
WebAssembly has no built-in I/O capabilities - it cannot access the DOM, read the filesystem, renders to the screen etc. In order to perform any of these tasks it needs to interoperate with the host environment (most typically JavaScript).
As a result, WebAssembly cannot detect its runtime environment. It could ask the host what environment it is executing within, although this could of course be faked!