The guideline was/is in the FAQs (helpcenter under on-topic):
We also encourage you to use your real name as your username. In your own enlightened self-interest, realise that participating in blogs, MathOverflow, the arXiv, and mathematical publishing are all forms of advertising for your "brand", even if that’s not your principal purpose (and hopefully it’s not). Since job applications require you to write your real name, you might as well use it everywhere else, too.
Using real names reminds everybody that they are corresponding with real people, and it demonstrates a certain level of personal investment in your MathOverflow identity. If you use a pseudonym and you get into some kind of trouble (e.g. fights in comment threads or spammy-looking posts), the moderators are much less likely to give you the benefit of the doubt.
But, it is also the case that this is really only an encouragement; a strong one, yet it is not a rule.
Moderators, at least some, are on record saying things at least as strong as they give less of a benefit of the doubt for users not using their real identity or are being somehow stricter towards them. And, I feel at least for some this also actually happening; to be clear, in my opinion, it is in no way excessive but still it is sometimes noticeable.
Multiple accounts are permitted as long as they do not interact in any way.
(For the idly curious: I do not have any.)
This was discussed many times and is a bit scattered, mainly on "tea."
A main, very old thread is Why not real names? (I believe, but was not active then, when it was created the practice and policy on real names was less clear.)