Praise be to Allah.
In Islam there are only two celebrations, namely Eid al-Fitr and Eid al-Adha; anything other than these two celebrations is an innovated festival or occasion and it is not permissible to celebrate it or to help others to do so.
The scholars of the Standing Committee said:
It is not permissible to celebrate the occasion of Laylat al-Qadr or any other night, or to commemorate other occasions, such as the night of the fifteenth of Sha‘baan (laylat an-nusf min Sha‘baan), the night of the Mi‘raaj (Prophet’s ascent to heaven), and the Prophet’s birthday (Mawlid), because these are innovations that have been introduced into Islam and that were not narrated from the Prophet (blessings and peace of Allah be upon him) or from his Companions. The Prophet (blessings and peace of Allah be upon him) said: “Whoever does an action that is not part of this matter of ours (i.e., Islam) will have it rejected.” It is not permissible to help others to hold these celebrations by means of money, gifts or distributing cups of tea, or by giving speeches and lectures on these occasions, because that comes under the heading of approving of them and encouraging them. Rather one should denounce them and not attend them."(Fatawa al-Lajnah ad-Daa’imah, 2/257-258 )
Based on that, it is not permissible for you to make cakes for these and similar occasions, because that comes under the heading of helping others to celebrate them. Celebrating them is not permissible, so helping others to do so is not permissible either, because of the general meaning of the words of Allah (interpretation of the meaning):
“Help you one another in Al‑Birr and At‑Taqwa (virtue, righteousness and piety); but do not help one another in sin and transgression. And fear Allah. Verily, Allah is Severe in punishment” [Al-Maa’idah 5:2].
The scholars of the Standing Committee also said:
Exchanging gifts is something that creates love and harmony among people, and dispels resentment and hatred. It is something that is encouraged in Islam; the Prophet (blessings and peace of Allah be upon him) used to accept gifts and reciprocate, and that is something that is commonly done among the Muslims, praise be to Allah. But if the gift is accompanied by a reason that is not acceptable in Islam, then it is not permissible, such as giving gifts on the day of 'Ashura or in Rajab, or on the occasion of birthdays and other innovated occasions, because that is helping in falsehood and participating in bid‘ah (innovation)."(Fatawa al-Lajnah ad-Daa’imah, 16/176 )
Making a cake and giving it to the child on this day, and telling him that this is his birthday so as to make him happy, comes under the heading of celebrating, even if his family do not call it a celebration. Undoubtedly celebration may take many different forms, but this does not mean that it is permissible. They have only chosen this day to give the child a gift because in their hearts they feel some love for this occasion and are trying not to let the day pass without celebrating it, even if that is only by giving a gift to the child.
And Allah knows best.