In most respects it's not a religious holiday, it's a public...

In most respects it's not a religious holiday, it's a public holiday, the Australian constitution includes separation of church and state, you don't get special privilege and an extra day off just because you have a particular belief. You get the day off even if you are not a Christian. Saying "happy holidays" is respectful because it works for everyone.

To be clear, I personally have no problem calling it Christmas, I'm a cultural Catholic, but even after dumping the religious beliefs I still greatly enjoy the secular aspect of Christmas which in my experience is the overwhelming bulk of the observance and has been for as long as I can remember. In my family we dragged ourselves to church on Christmas Day for years until finally we all just gave it up.

I also like to argue about Anzac Day, Australia Day, and the Melbourne Cup. Being hit on the head is next door. (Monty Python reference)