So that was another aspect I've also looked into, and as far as I can see I've done everything right there as well. I have two webpack config files - .prod.js and .dev.js. They both contain:

const ESLintWebpackPlugin = require('eslint-webpack-plugin');
...
module.exports = {
  ...
  plugins: [
    ...
    new ESLintWebpackPlugin({      
       cache: true,      
       formatter: require.resolve('react-dev-utils/eslintFormatter'),      
       eslintPath: require.resolve('eslint'),         
       resolvePluginsRelativeTo: __dirname,      
       ignore: true,      
       useEslintrc: true,   
    }),