My quick answer is "yes". But it really depends. For me, it's about the individual's objective for being on the web and joining a community in the first place. Both anonymity and "genuineness" have their own pros and cons depending on the type of community that one wishes to be in.
- If it's a business community for instance and your name is tied to a real one, having a handle diminishes the power of that name.
- If you're just using social media to look for cute people and maybe date them, it helps to keep you anonymous.
I guess the strongest and most compelling argument for anonymity is in special circumstances like political activism or other security related situations when your identity needs to be protected.