The problem is that Nationality is a relatively new term. To say that suggests nation states which didn't come into being until at earliest the 15th century and latest the 18th century. The idea of a hegemonic Greek identity, while tied to Helenic culture, doesn't preclude Greek nationalism. The concept of identity is tied more to regionalism and that of the city state before the rise of the nation state and even still exists to this day (which you see with regional separatist movements and even the identity of Southern in the US).

So, Alexander didn't have a "national" identity as it is anachronistic.