I suspect "recycling" there means something like "separated and sent to Thailand to fuel cement kilns".

This probably depends on the country, and perhaps even municipality, but plastic is recycled properly here (.nl).

Plastic, drinking cartons and metals are collected in the same container. These three are separated using magnets[1] and infrared scanners. The plastics are melted down to little pellets that can be used to create new products. A common example are shampoo bottles. This video is in Dutch, but it has some overview of these sorting machines: https://www.youtube.com/watch?v=Gf--8o3yD6g

Based on this video about plastic recycling in Germany, I think the situation is very similar there: https://www.youtube.com/watch?v=I_fUpP-hq3A The video is in English. The plastics that are recycled there seem to be PET, PP and HDPE.

I'm absolutely in love with all the neat little tricks these machines use to sort the stuff. From simple ones like sieves and centrifuges, to fans and advanced imaging stuff. It's beautiful.

[1] Even aluminium can be sorted with magnets by doing some magic.

I think one of the comments in here mentions that Japan only accepts PET plastic which apparently doesn't have these issues. Regardless the main issue is that the US is not strict about how we label things or regulate things related to recycling. There is only a small subset of plastics that are recyclable.