Illegimate kid's father

Let's say Y is a boy. His parents are not married. His father left his mother as soon as she gave birth to Y. Then Y's mother was punished for committing adultery and she repented to Allah and married a man. Can Y introduce his mom's husband as his dad? If not then what will he tell people if they ask about their father? According to Islam, you are not allowed to tell you are son of someone who isn't your father.