Well, the burden is on THEM to provide the proof to corroborate their claims, so you should ask them, or look for a reference in their email. If they aren’t willing to provide you with where they got their claim, I would consider it extremely suspect and would not believe it.
I tried to find some actual statistics but was unable to.
My intuition says a paper container is better because plastic takes a lot of effort to create and process, much more than producing wood does. Furthermore, plastic is generally not biodegradable, whereas paper is, thus in the long term I would consider paper to be better than plastic. However, it’s easier to reuse a plastic container (such as making it into a vase, or using it as a watering container) than it is to reuse a paper container.
I think the #1 best “low emission” mass milk delivery system is the one we have here in Canada. Our milk comes in 1.33L bags (3 bags come in the traditional ‘bag of milk’), stored in the fridge and put into a dispensing container when the previous bag is emptied. I think the amount of plastic is less than that used in a milk jug (for the same amount of milk), and the bags can be reused for many different things. Of course, reusable glass containers would be even better than this.