Hate these bubble envelopes! They cannot go into paper recycling because of the plastic liner, and cannot go into plastic film recycling because of the paper wrapping. The two are glued inseparably together and can only go IN THE TRASH!
I've seen padding made of something like cellulose foam, that the manufacturer claims dissolves harmlessly during the paper recycling process, and therefore they put the "recycle as cardboard" on the package. Those are rare though.
Yes. I got a package from Amazon US a week or so after this post and it had that written on it. I pulled it apart to investigate but the padding was not plastic bubbles like the ones in my OP photo. So it must be some sort of corn stuff that can be recycled, but unfortunately I've never had Australian Amazon packages in that recyclable padded packaging. We have a stiff cardboard bag that can be recycled but it's only single layer. My groceries did not even require padded packaging though!