All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His Slave and Messenger.
It is not permissible to give or accept a gift on the occasion of birthday celebrations, because exchanging gifts is a form of celebrating this occasion, which is not permissible.
However, it is permissible to eat from the food prepared for this occasion if the food itself is lawful. There is no reason to prohibit eating from that food, as preparing food for a birthday celebration is not sufficient evidence that eating it is prohibited.
Allaah Knows best.