Suppose I am refereeing a paper and wish to remain anonymous as referee. Suppose I had trouble understanding some of the proofs in the paper and made a post on MathOverflow and received a very helpful answer, that in my opinion will be useful not just to me as a referee, but also to the author in order to improve their paper. It seems reasonable that when I point out the argument I received from MO I should acknowledge where I got it from. But in doing so I will need to point to my post, and therefore reveal my identity. Would it be appropriate in this scenario to make the MO post using an anonymous login, specially made just for that purpose? I would be interested in hearing what other ways to proceed may be.
I don't think there is anything inappropriate about using alternate accounts to ask questions. I have asked many questions using alternate accounts over the years (for all kinds of reasons, e.g. shame at not knowing something basic!). As long as your different accounts don't interact (and especially don't vote for each other!), this does not violate any rule that I am aware of.
That being said, I think it is a little weird to initiate an anonymous online discussion of a paper you're refereeing. I'm not sure I would go so far as to say it is unethical, but it's definitely not standard practice, and I think that many editors would get annoyed at one of their referees doing it.
On my opinion, if you have a question on the paper that you referee, you may ask only the author. (You can hide your identity if you wish by passing your question through the editor.) A referee is not only anonymous but also is supposed to be confidential. Which means you are not supposed to discuss anything in the paper that you referee with the third parties, anonymously or not.