This is expanded polythene foam (EPE). Also known as 'JiffyFoam' (as in 'JiffyBags', the padded envelopes).
I've been using the stuff as lightweight sit/casualty mat for years, since encountering it during my industrial year as a student (1984); it was used as electronics packaging. I carry a piece of 3mm thick and 35cmx55cm which weighs 14g. Since it's anti-static, it's a fetching shade of pink.
I use some thinner (2mm?) sheet as a nice insulating groundsheet when car camping, and sometimes an even thinner 1mm sheet when walking. The tiny amount of insulation is gives does seem to warm the tent, and reduce groundsheet condensation.
The 3mm sheet gives remarkable insulation for the weight, IME. The insulation comes from the fact that it's made up of lots of tiny bubbles, so air cannot circulate within the thickness, unlike bubble wrap. I've used the insulation to play around with making insulated cups from drinks cans (e.g. a 355ml Red Bull can inside a 500ml can, with a layer of 3mm EPE foam slipped between).
The packaging grades have a tougher surface than the laminate floor underlay stuff, which is weaker, and tears easily.
It's waterproof and non-absorbent.
It doesn't give much cushioning, even at 3mm, but it does insulate well; hence the casmat application (assuming you can get the casualty onto it...). It won't give much puncture protection, either, especially the 1/32" you're looking at.
It's also quite bulky to pack, as it doesn't compress.
I've always salvaged the stuff from bins or skips.
You can also get it in big chunks, often used as a replacement for expanded polystyrene foam for packaging things like TVs and monitors. I use this quite a lot for making bespoke protective packing, as the foam cuts easily with a sharp craft knife. My ice axes have nice head and spike protectors for packing in my duffle when flying. I've even used it to make a custom-shaped mouse when I first started getting RSI... Likewise, my skis are protected in their bag using chunks of this foam.