What does "personally identifiable" mean

Question

I was reading a booklet when I came across the phrase "personally identifiable". What does it mean in the following context (emphasis mine):

As the common household and business router will often display just one IP address for all of the people connected to it, the IP address will identify a group of people rather than just one individual. As a result, it is often hard, if not impossible, to be sure who exactly did what, purely on the basis of an IP address.

On the other hand, IP addresses are very often personally identifiable, and so, following a basic precautionary principle, must be treated as such unless they can definitively proven not to be.

First paragraph says that it's not easy ("often hard, if not impossible") to identify the exact person who did something behind a router shared with others. But "personally identifiable" in the second paragraph seem to say the opposite. I think I'm missing a nuance here.

Update: The booklet can be downloaded from http://www.edri.org/files/2012EDRiPapers/how_the_internet_works.pdf It does not define "personally identifiable".

Answer 1

The two paragraphs mention two cases that often occur: one where an IP address does not reveal personal identities, and one where it does. The text probably discusses both cases to lessen the likelihood that a reader will assume that either case alone accounts for nearly all situations. Note, the first case is good if one wants to avoid infringing privacy laws while collecting IP numbers, and bad if one is trying to find out who posted something scurrilous; and vice versa for the other case.

Edit note: The following is based on and expands upon the first paragraph of my original answer.

As noted in a comment, the meaning of personally identifiable is related to the concept of personally identifiable information, that is, to information that may be covered by privacy laws and information security concerns.

The booklet mentioned in the question, “How the internet works”, by McNamee, Fiedler, Humeau, and Maisuradze at EDRI.ORG, apparently isn't in aid of specific interest groups such as privacy advocates, data thieves, governments, businesses, etc. The section at issue (Page 5, THE IP‑ADDRESS) neutrally presents general information. It apparently is intended to slightly inform Internet users who don't know what an IP address is. As such it is worded somewhat informally and diffusely but appears to be entirely and technically correct.

Just before the paragraphs quoted in the question, the booklet says “In many cases, IP addresses may be used to identify an organisation or individual that has acquired the services of an Internet Service Provider... the IP address does not always identify the person behind some digitally traceable act”. This context suggests that the phrase personally identifiable as quoted in the question means traceable to an individual.