I recently came across this story in Packaging Digest, which is called "Are Paper Bottles Sustainable and Practical?" It features an interview with Lux Research Associate Drishti Masand. 

It addresses several of the issues I've been thinking about with paper bottles, as a few brands have announced they'll be moving some of their packaging to paper bottles. 

We'll start with his conclusion: 

Many companies are adopting solutions that solve the issue of plastics waste; however, they create a new set of sustainability challenges. All the alternative materials solutions have a separate set of problems associated with them; thus, they simply shift the packaging problem. With that, paper packaging will continue to grow for adoption, but will never be a substantial threat to plastics.

I thought this was super interesting:

However, current available data suggests that paper packaging generally requires several times more mass to fulfil the same function as its plastic counterpart. As a result, the overall environmental impact tends to be higher for paper, except in its carbon footprint. Additionally, replacing plastic with paper could lead to a serious supply problem. Paper is a short-term solution and will simply shift the burden for packaging problems.

And this is something I suspected - If you need a plastic insert, why not put the alcohol in a plastic bottle? 

Paper bottles need a coating or plastic liner on the inside to provide moisture barrier, and resistance to other environmental factors. While companies claim the layers can be easily separated for recyclability, we are skeptical of those claims given the challenges in recycling plastic-lined paper today and the likelihood that the companies aren't using any technology to allow for that separation.

Also, the cap and closure for these paper bottles is aluminum- or plastic-based, so it would need to be separated and sent into different recycling streams — that is highly dependent on efficient collection and sorting, and so again makes us skeptical of the end-of-life processing success rate.

So, as in the case with aluminum versus plastic versus glass, there are pros and cons to each. At least with the paper bottle there's less bottle weight when compared to glass.