First, let's split this into two areas: public anonymity and internet anonymity.
Public anonymity is for people who don't want the world to know who they are. Don't judge them for not wanting to be 'famous' as there will be thousands of people with different legitimate reasons, perhaps not wanting to be bullied for their opinions or not wanting to openly discuss something they can't come to terms with from their past e.g. they may have been abused and want to give support to other abused people without public admission of their own embarrassing secret.
Internet anonymity is where people sign up for a service, such as email, and don't want to register all their irrelevant personal details as this could be open to exploitation. It's bad enough that large service providers, such as google and
facebook, monitor browsing habits and feed through adverts from the
browsed pages or similar.
Yes it's important for people to remain anonymous, for hundreds of reasons. But it's also important to protect users from idiots (a collective term for anyone who persistently uses the internet for illegal, immoral, uncivil or just plain nasty abuse). I say persistently because anyone can make a mistake or two and such errors of judgement should be treated accordingly, with an apology and good behaviour forever after being a suitable means of punishment.
I'm not a god or the brainiest person on the planet so I don't have an answer for how to sort it all out. There are even levels of acceptable behaviour from one person to the next: some people are offended by swearing, others by bad grammar. Tips: on social media, choose contacts wisely, and where there is an opportunity of moderating content, use it.
Whatever the answer is, let's try to work it out without 'Big Brother' restricting what little freedom an ordinary person has.