How to ask a Web3 RPC client which chain it is working with?

You need to create a web3 instance. for this you need a provider:

const NETWORKS = {
  1: "Ethereum Main Network",
  3: "Ropsten Test Network",
  4: "Rinkeby Test Network",
  5: "Goerli Test Network",
  42: "Kovan Test Network",
  56: "Binance Smart Chain",
  1337: "Ganache",
  137: "Polygon",
};

// you need those 2 npm packages
import detectEthereumProvider from "@metamask/detect-provider";
import Web3 from "web3";

const provider = await detectEthereumProvider();
// Only if you have a provider then create a web3 instance
 if (provider) {
        const web3 = new Web3(provider);
        const chainId = await web3.eth.getChainId();
        if (!chainId) {
        throw new Error("Cannot retreive network");
        return NETWORKS[chainId];

    }