AIBU about gifts for the non birthday child?

[Madeaminnieme]

It's my DS first birthday coming up and I'm taking my 4 year old DD shopping for a gift to give to her wee brother. My husband wants to get something for her too but I've said no because I know the grand parents are doing that and I want her to know that birthdays are special for the birthday child. This all stems from me as a child asking for a gift and getting shot down quite harshly so fear she gets used to it then when it stops, feels more put out.
If the grandparents are doing it (and likely to still spend a fortune on her) AIBU to say no from us? Or is it the done thing?

We do get the non birthday child something small, just the same as I did when it was my siblings' birthdays. None of us are spoilt or expect presents on friends/sibling birthdays now.

Only the birthday child here gets gifts. However other child gets cake, special tea, balloons and special outing too so still benefits from their siblings birthday.

My exDP endured this! His parents would buy his brother a present on DP birthday and even so far as buying brother something DP wanted. Wrong on so many levels.
A birthday is the one day of the year that is all about you and if you can't bring up kids who understand that then something is wrong in the world.
In my family it wouldn't have even occurred to my mum to buy my siblings gifts to 'compensate' or 'soften the blow' whatever- every child has their own day so I don't see the need and actually see it as a ridiculous indulgence.

My DH said he grew up getting gifts when it was his brothers birthday so doesn't see the issue. I just feel I am constantly having to watch what I say round the in laws because I erupted after they took my DD to see Santa before we did a few years ago and was told I over reacted to that. (And I still get angry about it too)

You don't need anything for the non birthday child. They can get a lot of pleasure getting involved in present buying for the birthday child. They get their turn. I had never heard of this odd idea before MN.