There is actually a bit of sense in there. Paper cups weren't simply paper - its tetrapak-like material with plastic coating inside. They are notoriously hard to recycle. Plain plastic cups on the other hand are made from single material, most likely PET. Moreover, they are transparent, without colouring additives.
There are two reasons why colour in plastic makes it harder to recycle. First, pigment is a completely different substance, which behaves differently from plastic itself. It makes it harder to "re-melt" into stable material. If you ever 3d printed anything with matte/gloss filament, you'll know that it is more finicky than plain one. Second, uncoloured plastic can be coloured into anything, while other needs to be either sorted by colour or mixed with strong dye (black, gray, dark brown, etc) to have consistent colour.
PET itself is pretty easy to turn into something new, actually. A workshop near me had a live demo of the whole process - chipping it into small pieces, feeding to the heated tube, and then injection molded into trinkets. Industrial grade processing usually have "turned into pellets" step in between, but it's basically the same.
Plastic-covered paper, on the other hand, should be somehow separated first, and then handled with two different approaches - one for paper, another one for plastic film. Doable, but much harder. Paper straw can probably decompose by itself, without any special conditions.
UPD: Be wary that recycling is not a panacea. There's multiple videos about how recycling plastic isn't actually a thing. https://www.youtube.com/watch?v=68zjxTTl5Ik for example.
