Usage of "Which Birthday"

I can see both points of view.

'Birthday' is used as shorthand for 'annual commemoration of your day of birth'.

In this sense, you could say it implies a choice of options, including, say, '33rd birthday', '50th birthday', '99th birthday', etc.

Strictly speaking, though, we can only ever have one birthday - the day on which we were born.

So, paradoxically, your 20th birthday (in the first sense) would actually be the 19th commemoration of your actual day of birth. Put that to someone born on 29th February, or Queen Elizabeth II, who has 2 birthday commemorations every year, and you're in even more murky natal waters.

While it could be argued that, "Which birthday?" is grammatically correct, a native speaker would generally ask, "How old are you?"