Wa `alaykum As-Salamu wa Rahmatullahi wa Barakatuh

There is nothing wrong in wishing someone with such words: “Happy birthday” “many happy returns of the day” etc. Allah says: “AND LO! We accepted this solemn pledge from [you,] ‘ the children of Israel: “You shall worship none but God; and you shall do good unto your parents and kinsfolk, and the orphans, and the poor; and you shall speak unto all people in a kindly way; and you shall be constant in prayer; and you shall spend in charity. ”And yet, save for a few of you, you turned away: for you are obstinate folk!” (Qur’an: 2: 83)

The order to speak to all people in a kindly way means that we should follow the customary words of greetings and reciprocate kindness with kindness. The only exception is when such words imply paganism or aggression or sins.

As for the question whether we are allowed to celebrate birthdays,  I would like to cite my earlier answer:

I do not find any reason to consider birthday celebrations as haram as long as they are free of any forbidden or undesirable practices. It should be judged as we judge any other customs; customs, by themselves, cannot be deemed as forbidden, unless they contravene any of the stated principles. Unlike acts of worship, the original rule in regard to customs is permission; for humans are free to do so unless otherwise forbidden.

Humans are allowed to thank and celebrate on important occasions in the lives of individuals or communities. Therefore, there is nothing wrong for individuals to celebrate the day of their birth; however, while doing so one should steer clear of unnecessary wastage or activities forbidden in Islam.

Almighty Allah knows best.