I think you’re focusing on what really amounts to a bad nickname for the generation that obviously is Generation Y. (Gen X, Millennials, Gen Z, I wonder what letter was left out??)
Secondly, a millennium is a thousand years. Are you saying the previous thousand years (1000-1999) don’t count as a millennium that millennials… existed in?
Thirdly, it’s the change from one new millennium to another that people were excited about, no one gives a shit about the before or after. It’s simply excitement about the changeover. In 2024, no one gives a shit that we’re living in the “new millennium.” The song goes “let’s party like it’s 1999” not “let’s party like its 2001” or “let’s party like it’s 1981.”
Finally, last I checked, humans tend to celebrate things before they come to pass, kind of like how walking for graduation comes before finals. We celebrate New Years Eve all night leading up to the New Year. New Years is over when the new year has actually begun. Nobody celebrates on January 1st.
So literally no one born in the new millennium gives a shit about it being a new millennium. Only people born before it cared or would care.