Praise be to Allaah.
If this food is some of the Shi’ah food that they make on ‘Ashoora’ along with the cheek-slapping and hitting themselves that they do, then it is a reprehensible innovation that the Muslim must keep away from and not participate in it. We have quoted the fatwa of Shaykh ‘Abd al-‘Azeez ibn Baaz (may Allaah have mercy on him) in the answer to question no. 102885. But if this food is not accompanied by those things and the aim is just to treat oneself and one’s family, then there is nothing wrong with that and it is not described as a bid’ah (innovation).
Many scholars have stated that one should treat oneself and one’s family on the day of ‘Ashoora’, and several ahaadeeth have been narrated from the Prophet (peace and blessings of Allaah be upon him) concerning that, but they are all da’eef (weak) and are not saheeh (sound).
Celebrating birthdays is a reprehensible innovation. This has been discussed previously in the answer to question no. 1027.
Eating food treats on that day and bringing sweets and fruits is a kind of celebration and veneration of that day, and should not be done.