In essence, the authors state that, the closest thing you can have to "identity" in the crypto-world is your public key (or equivalently, its hashed version, which is otherwise known as bitcoin address).
This is certainly nothing new to the more experienced members of this forum, but it can be somewhat confusing for the newbie users.
If you are a beginner, you already know that you have a username and, in addition, you have used an email address to register on the forum. So, at first glance, it might be logical that your identity should be tied to your email address.
However, it is NOT. Don't forget that this is a cryptocurrency forum. There is a better and more secure way in which you can prove your identity. And, it is not connected to your email address.
It has to do with a bitcoin address you own and you have control over. All you have to do is to publish it somewhere (in one of your unedited posts).
In case you need to prove that you are indeed the person who owns your account (for example, in case your account gets hacked), the administrator of this forum requires proof of ownership of that previously published bitcoin address.
In fact, on BCT you can't use your email address to prove your identity even if you wanted to. This inability to prove identity via email address is, surprisingly, something even some of the more experienced users whose accounts have been hacked complain about (from time to time).
But, as already said, your identity is your public key, and I'm not surprised that the administrator of this forum insists on it.
Let's get a little bit more into identities in the crypto space.
If you want to speak to the world on behalf of a certain identity (public key), you have to possess the private key that corresponds to that identity.
This makes a lot of sense, because you can then sign a message by using the corresponding private key and share that message with the world. The world can then verify that it is indeed you, the identity behind that public key, by verifying the signature.
So, again, in the crypto-world, your identity is your public key. Moreover, you can have a lot of public keys (aka bitcoin addresses), and this has as a consequence that you can create numerous identities.
You can use a certain identity for a couple of days and decide to switch to another identity after that.
In the real world, your identity (your identity document) has to be managed by a central authority. And this identity is permanent, or at least not so easy to change.
In the crypto world, your identity is managed by you in a decentralized way, and you can change it as many times as you like.
This loose sense of identity doesn't mean that you are totally anonymous or that you have total privacy. The reason is that your crypto identity leaves traces and behavioral patterns that can be traced and can lead back to your real-world persona.
You can read more about privacy/anonymity in bitcoin here.
Now, because your identity is essentially a random 26 to 35 alphanumeric sequence, you might be wondering if another person could possibly assume the same identity as yours (the technical term is collision). Even if you change your identity 1000 times every day, chances of this happening are negligible. In fact, here is a nice illustration of how impossibly difficult a collision with another bitcoin address is:
In conclusion, if you are a beginner, to be able to verify your identity in case of unforeseen circumstances, do the following:
1) learn how to sign a message with your bitcoin address from this excellent tutorial.
2) sign a message of a predefined format (as explained in step 1) and post it here.
Now you can then be sure that you can prove your identity in case such a proof is needed.
Edit: Revised according to the corrections indicated in @pooya87's post below.
What is the different between public address and public, because I open one online account and I see something like public key, private key and public address are them the same?