How to override / customize existing Widgets?

Generally, no. Many widgets have action/ filter hooks where you can override certain pieces, but Widgets generally aren’t like Themes which can be overidden with Child Themes.

Some widgets are written in such a way that they can be overridden, but that’d be on a case by case basis. Something like this you could declare the Class in an earlier hook and “override” the original.

if(!class_exists('My_Widget')) {
    class My_Widget extends WP_Widget {
       // ..

You’d need to read through the source code of the widget in question and see what options the author provides (or find the documentation if that’s any good). Here’s the source for WP’s Recent Post widget as an example.

Most likely your best bet is to make a new Widget to suit your needs, possibly extending the original or copy/paste it as a starting point if needed.