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!
It honestly depends on your local municipality. Where I live they don't take anything but straight up hard plastic that is cleaned, and clean cardboard, cleaned tin cans. Anything else they don't process. And even then they're probably throwing out the whole truck of recyclables cause one dingus didn't clean out a can and it got tomato soup on everything else. Recycling is honestly a scam in America, especially once China stopped buying our waste years ago.
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!