Eventhough JavaScript was first developed to be used in a browser, the language itself has never been a strict client side language.
Aside from a few string methods that create HTML tags, there is nothing in the language itself that is specific to client side programming. Everything related to the browser is in the DOM, not in the JavaScript language.
JavaScript is used in client side scripting (and that use is almost analogous to client side scripting), but branding it a client side language would be misleading. The fact that the language itself doesn't contain the browser integration has made it easy to adapt it to other environments, including server side scripting.
Node.js is far from the only server side scripting using JavaScript. ASP for example offered VBScript and JScript (Microsofts implementation of JavaScript) as server side languages. This list compares 28 different server side uses, and it's not all there is.