I think the point is that it's not a printed medium. You can do anything you want with paper... It doesn't give restrictions to what you want to do. You can doodle, make a mind map, sketch an amazingly detailed human eye, or actually take notes and brainstorm in your meeting. Paper is flexible. Software is not. They think they've made a good attempt with "Paper" at providing that flexibility online.
Chrome, the browser, was born with the idea of reducing the amount of "chrome" (frames, controls and other UI elements) in the app. I guess developers like to have fun when naming their projects.
It was, more specifically, the online equivalent of a face book [0]. Which implies an answer to why name a product after printed media when it is not for print -- because it serves as an online equivalent or replacement.
However, a face book is a highly specific thing, where as 'paper' is about as generic as you can get - even wasps make it. Can I mop up a spill with dropbox paper? Make a hat out of it? Cut out a coupon? Make a saw[1] out of it?
Not to mention that it's far from the first product named 'paper' in our tech bubble.
Pretty weird to name it after a printed medium if it is not primarily meant for printed media.