Some carrier bags are made of a plastic marked 2, gardenoma, so you would be safe to take your air pillows to the supermarket. I started recycling other plastic bags, like the sort that magazines come in, with the very few carrier bags I couldn't avoid, ages ago. It had occurred to me that the bags didn't know that they didn't have handles and couldn't be described as carrier bags, but that wasn't their fault and it was a shame to discriminate against them. Nothing wrong with stopping using Amazon though!
Some plastic wrapping (an increasing amount, and including some fruit net bags, I am happy to say) is marked "Recycle with bags at larger supermarkets. Do not put in kerbside recycling" or words to that effect. That will be why yours are being left, 25Avalon, I would imagine.
Hope that helps.