I think the birthdays are exactly the same for mayors and non-mayors. The only difference is the friendships between the player and villagers. If you haven't talked to villagers much on a certain character, you won't get a party
Maybe your friendship level isn't high enough with any of your villagers? If you never talk to any of them with that character, they won't have a party for you.
If the friendship level is high, there actually are some birthday-only items. All of them are non-orderable. They include: