Does Written Torah allow birthdays and birthday's celebrations? As far as I know, there is no single reference to a birthday of a Hebrew in the Written Torah, except the birthday of Pharaoh. Let's take a look. It is Genesis 40:20
YLT Gn 40:20 And it cometh to pass, on the third day, Pharaoh's birthday, that he maketh a banquet to all his servants, and lifteth up the head of the chief of the butlers, and the head of the chief of the bakers among his servants,
As you can see from this verse, a birthday celebration was clearly practiced in Egypt and was definitely the Egyptian custom. But what about Hebrews?
I know of only one clue that can support a claim that Hebrews celebrate birthdays too. It is Genesis 21:8.
YLT Gn 21:8 And the lad groweth, and is weaned, and Abraham maketh a great banquet in the day of Isaac's being weaned;
Obviously, if Abraham celebrated weaning of Isaac, he most certainly celebrated day of his birthday as well. Isaac was beloved son of Abraham's old age, so he was dear to him, meaning that he celebrated everything about his son, including his birthday.
A birth of a child, particularly sons, was a very important occasion among Hebrew people. It is Genesis 22:20
YLT Gn 22:20 And it cometh to pass after these things that it is declared to Abraham, saying, 'Lo, Milcah hath borne, even she, sons to Nahor thy brother:
As you can see, a birth of a child was a good news that was delivered to Abraham.
Not to mention, that we have extensive genealogies in the Torah that emphasise how long people lived, which means that counting years of one's life was important to the Hebrews.
So I think we can definitevely say that a birthday is a good tradition that we should embrace. There is nothing selfish or abhorrent to God in this tradition. So I think Torah definitely allows it, even though it definitely does not command it anywhere.