Out of the box, when choosing options, images, price, etc will change, but not SKU.
In order to display the SKU of the selected product, you can create a Plugin (Interceptor) for Magento\ConfigurableProduct\Block\Product\View\Type\Configurable::getJsonConfig()
where you need to add missing SKU
for each simple product.
Switching SKU
when selecting an option can be implemented in vendor/magento/module-swatches/view/base/web/js/swatch-renderer.js
and vendor/magento/module-configurable-product/view/frontend/web/js/configurable.js
depending on the type of attribute.
I made free VCT Change SKU Dynamically
module that solve this problem.