I want my baby to have his own bday

So my due date was changed and as an excited mom who gets to meet her baby sooner than expected I posted to fb to share with family and friends. Well then everyone with an October birthday starts commenting "oh you should have him on my bday" "wouldn't it be cool if our kids shared a bday" Nooo.... no it wouldn't because than he wouldn't have his own day to celebrate. I mean I understand he isn't gonna be the "ONLY" person with that bday but within the family I would really like him to be. Am I overreacting? What are some of yalls opinions on the matter?