I understand the Jewish birthday is calculated on the lunar year, which could cause the birthday to drift back and fourth along the lunar calendar. But supposing the child was born, say, first day of Sukkot, would his birthday be forever tied to that holiday to make calculations easier?
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Yes, someone born on a certain Hebrew-calendar date will have the same calendar date as his birthday every year. Thus, some one born on a red-letter day that is the same date every year, like the first day of Sukos, will have that as his birthday. (In fact, I have seen yahrzeit plaques that indicate "1st of Sukos" or the like instead of a date.)
Some red-letter dates vary, however, like most fasts (which change date if they occur on Shabas), Shavuos (which nowadays is the 6th of Sivan but won't always be), and some others. The birthday follows the calendar date, not the red-letter day.
Exceptions to the above rule show up
For those cases, see e.g. http://download.yutorah.org/2013/1053/Purim_To_Go_-_5771_Rabbi_Willig.pdf.