i really do not understand where this idea that plastic has something to do with the climate came from, how do people imagine that to work?
No, the point of not using plastic is to not have plastics blowing around on the street for 50 years before it's degraded into microplastics that instead enter our bodies.
Some materials have higher carbon emissions than others, in terms of refinement, processing, and transportation. The third point is location dependant, but creating and shaping different materials will have different contributions to global warming.
Edit: There are also concerns with the product's end of lifespan. How long it takes to biodegrades, how easily recyclable it is, and how much the available disposal methods will effect the environment. Plastic is not great on several of these accounts. Recycling plastic water bottles isn't very efficient either, compared to glass bottles for instance.