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.
[0] https://en.m.wikipedia.org/wiki/Face_book