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.