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?

I agree with that the comment you're replying to is basically nonsense, but I do have two points to correct about this.

First, a small nitpick. Technically, millennia go from 01–00, so 1001–2000, with 2001 being the first year of the new millennium.

More significantly, it is obviously the case that millennials were so named because of something to do with the turn of the millennium. Frankly I don't know what that is and it would have made more sense to name gen Z millennials because they actually span across the millenium divide and are the first generation born into the new millennium. Or if gen Y had started and finished 5 years later, they could have spanned the bridge, as well as even older genYers still being children during it, which would have been more appropriate.