From what I've learned and tested (after witnessing a friend's celebration in-game), receiving the exclusive birthday items depends on your friendship level with certain villagers. If you're not at a certain friendship with them, such as recently having moved them in, they won't give you any birthday items, period.
If everyone's the same level, three villagers will inevitably have to pick up the slack and give regular items instead.
(because of the +1 at the party, there are 8 items total IIRC)
So from this point on, it's assumed/hoped that the villagers in question have hit that threshold and are able to give you birthday items!
For the three at your party, the only way you can get the items is through the daily gift (with the cupcakes in your inventory, it will only highlight them).
Even though the 'host' gave you one already, they're counted in on this, too!
I'm pretty sure sharing a birthday won't completely lock you out of it, it just requires some timing and being on at the right hour.
If they have the same birthday as you, it should be that they start their own party once you're put back in front of your house after your own.
(so this means they can attend yours, too)
I really don't trust the idea of giving one's birthday twin a cupcake *as* their present, but I'll test it out anyways to see what happens! I think it's just coded as a regular gift, but I could be wrong.
IIRC, their parties end at midnight, and that is when you can give them a daily gift.
This is when they (should, if they weren't at your party) wish you happy birthday (because theirs is 'over'), and you can give them the cupcake, et cetera.
Now if they go to bed immediately after their party, my only other idea is: villagers give you belated birthday wishes if you didn't talk to them on your birthday. This could segue into the cupcake dialogue *or* it's the same as the party trio: daily gift substitute.
My friend didn't give out all the cupcakes on their birthday (as they didn't realise the party trio were eligible), but were still able to receive the last items the day after.