You can add your copyright statement in whatever form you like - it can be with your name, your pseudonym, both or completely skip it. You choose what or if anything at all you want to have written there.
The purpose of the copyright statement is to indicate the authorship of the code. This allows other people who re-use your code based on the license to attribute and acknowledge you.
Further and conversely, if you don't add a notice which in some way identifies you, it might be later on harder to claim that it was your work, as will it be harder to judge if some other project uses without license code from your project.