Babel, via babel-loader, transpiles differences in ES6 syntax. Babel on its own does absolutely nothing to add in ES6 standard library functionality (like Object.assign). Loading the polyfill loads a separate polyfill core-js for you, but you can load any polyfill you want.
Even some syntax conversions rely on specific polyfill functionality to be loads, since some syntax relies on algorithms and behaviors implemented in library code. The ES6 features on http://babeljs.io/docs/learn-es2015/ each list what standard library functionality are assumed to have been loaded.