Lash said Circumstances of his birth, more so than just the date IIRC. We can't consider the year as significant, not yet anyway, since even JB is having a hard time deciding on his year of birth. We do know his date of birth, who his parents were, possibly even approximate time of birth if my own theory holds based on events in BR and how Maggie died either while giving birth or just after.
We don't yet know for certain what other Circumstances may have played into this, but I believe that there must have been others.
Well, we know from CD that on Harry's birthday the immortals reacquire (or fight for, or whatever) the power they wield. I would guess that Harry being born on Halloween plus some other circumstances around said birth resulted in the Starborn status.