It doesn't break the message to lack identity. It forces you to consider the message on its own, stripped of whatever cultural context or baggage you might attach to it.
Though it can take practice, we can and do make use of anonymous messages. It is merely a new "mode" of communication, and it gives rise to certain types of communities. However, anonymity must be consensual for it to work. Law enforcement will still come after your identity depending on what you say, and that is a case where the consent is broken (or never acknowledged according to them). Messages become drive-by shootings, and this has an effect on the exchange.
Also, user names are not the equivalent of anonymity. On HN, karma is a feature of each identity, and baggage can be attached just as easily. And if someone says anything interesting, we can read their other comments to learn more about them. As we read more of what they've said, their user name will begin to have a unique aura. Now if we happen to read a new comment and recognize the name, it will ring a bell automatically, and we will already be working with more information than what was included in the utterance alone. At this point, we are not forced to consider the message on its own. We are working with identities that just may or may not be real.
It's kind of a mess, and it's really not fair to anyone.
Anyway. Yes, anonymity carries its own baggage. But it's a separate sort of baggage, out there to be assumed by anyone at any time. There's a long and proud history of anonymous authors affecting politics or writing biting satire that would have been impossible without anonymity.
'Identity' really isn't necessary, anyone and everyone can see the logic in something (or perhaps lack thereof). I guess what I'm saying is identity is tied to things that are often controversial like sex or race, but also issues that should be universal (like respect for all races and sexes), but it doesn't take an identity to make those things true, those things are true because (hopefully) we believe people deserve to live and live as they chose.
Identity is mostly shaped by reputation and by impression, both of which are not in our full control, and both of which just happen. So there isn't anything to believe. Identity is already part of every message, and the lack of identity is already consequential, unless one deliberately dismisses those consequences. At this point, anonymous exchanges require skill, and this is a new skill that, for example, my parents would not ever be comfortable with. The idea that "no one is talking" is new and quite novel actually. And has only become possible at large scales (as communities) with the internet.
This is why when communication is anonymous, not only does it change the message, it also breaks it in many ways. If we hear Obama say something, we will hear something else than if Trump said it, even if the words are exactly the same. And what if either said something anonymously? We would certainly care less, if at all.
It is only natural to consider the source of the utterance together with the utterance, and vice versa. When we speak, we skip a lot of what is included in our identity and the context we bring along with it. The kicker being, even if our identity is not part of the message, the message is still signed with it. We think of you when you speak, even if you're talking about someone else.
Not quite anonymity, but body language and tone also provides bandwidth. That is why emails can be dangerous when sent to someone who infers emotions from words that would otherwise not have been communicated in person.
Communication is complicated.