score:0

final update in webpack.config.js

test: /\.(js|jsx)$/,
exclude: /node_modules/,
use: {
     loader: 'babel-loader',
     options: {
         cachedirectory: true,
         presets: ['react', 'es2015', 'stage-3']
     }
}

score:1

as pointed out in a comment there has to be something with babel configuration that webpack can't transpile jsx syntax.

check out if you have babel-preset-react installed and turned on:

"presets": [
   "react"
],

little nitpick - use caching for faster incremental builds:

{
   test: /\.jsx?$/,
   exclude: /node_modules/,
   use: {
     loader: 'babel-loader',
     options: {
       cachedirectory: true,
     },
   },
},

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