No he is pointing out irony. He is saying that a lot of people feel like they should be anonymous, but that others, fit whatever reason shouldn't be allowed to. Celebrities, pundits etc.

It's akin to celebs complaining about gun control, stating they hate guns and would never have one themselves, as they are guarded by several armed body guards.

No, I don't understand why people want anonymity while also demanding that people can't anonymously send them messages on a public platform the entire point of which is the sending of messages.

People want to be anonymous, but don't want just anyone to be able to send them messages. Okay, so who can send you messages? Who can post things you can see? What is your solution to this?