I am fairly certain the reason all horses have one 'birthday' is to even out competition to a point. In America, it's January 1st and for the Triple Crown races, horses have to be 3. Most breeders try to breed for as early in the year as possible (usually using hormones to trick mares into thinking they're in heat so they can be bred), so that the horse is actually very close to what the rules say its age is.
It wouldn't be entirely fair to have a horse born in February vs a horse born in August who would both be considered 3 on the same day. The August foal won't have nearly as much experience because they had to wait 7 or 8 months AFTER the February foal has been started on.
It's probably different in Australia because your seasons are backwards from ours, so by putting it in August, the season would probably be the same as if it were January here.
However, that's usually only for competition, particularly racing. I wouldn't celebrate my mare's birthday on January 1st because her's is June 26th.