I'll admit I've entertained the thought myself, usually when Youtube's involved. However, there are several reasons why it wouldn't work:
1) I've seen people post the most horrendous comments on various articles and such using their facebook profiles. Not just on facebook itself where they can control the audience, but on all sorts of other, very public sites with facebook plug-ins.
2) As JoJo said, it would make it much easier for people to get really nasty with threats against family etc.
3) There's no way to enforce it. What's to stop people from making fake profiles with vaguely realistic-looking information?
4) Anonymity has its upsides as well as its downsides: some people are more comfortable discussing all sorts of sensitive topics anonymously. For example, someone who's gay but not out to their family can express themselves behind a pseudonym. Someone with a mental illness can talk about it frankly without fear of an unsympathetic employer linking the comments back to them. People can ask questions about embarrassing medical issues they've been having. You're free to vent about people in your life and their behaviour (irritating family, inept colleagues, creepy exes...). The list goes on.
5) Sometimes you just don't want people you know IRL being able to track down your every profile. I don't post anything I'm ashamed of on here, and I'm pretty free with my information, I've linked to stuff with my real name on before now for example, but I'd still be a bit weirded out if one of my RL friends or family realised this was me and started checking out my posting history etc. I've abandoned forums before because I realised other people from my college were on there and I didn't want them to identify me, again not because I'd done anything wrong or embarrassing, but just because I didn't want to mix those aspects of my life. It's kind of like not wanting relatives as facebook friends (all mine are blocked) - sometimes you just want to keep different parts of your life separate.