Anonymous betting means that each of your trades would appear as "Anonymous" in the trade log and not be directly traceable to your account.
Anonymity may be useful in betting against things that are socially desirable (e.g. betting against your friend's chances of success in business, dating, etc.).
I agree, people are allowed a pseudonymous alt which puts the ball in their court for anonymity. A dedicated manaccountant could still piece together a large anonymous bet with someone's profile to figure out who it was, so this would still be worse than using your alt for anonymity.
I think this would mostly be used for poorly disguised profit transfers, of which there are already too many.
I think this would make the site a lot less fun. I love seeing someone make a big bet and then going to their profile and seeing what else they're betting on and if there's something big happening. It would be optimal to always be anonymous if the option was available, because there's no reason to give information to your adversaries. It would turn into a big prisoner's dilemma, where the site would be more fun and accurate if no one did it but everyone has an incentive to defect and be anonymous while hoping no one else does so.