Is it permissible to buy cakes and toys for children on Eid al-Fitr and Eid al-Adha to bring joy to them and compensate them for not celebrating their forbidden birthdays, especially given the absence of Eid manifestations in the country of residence?

The Eid days for Muslims are Eid al-Fitr and Eid al-Adha. These are days of joy, eating, drinking, and generosity. There is no harm in making cakes and foods for these occasions, as preparing special foods is one of the manifestations of Eid, and it is not an imitation of non-Muslims.