I once read a fascinating article on this topic. There was a case a few years ago in Portland, Oregon about someone who confided in her Rebbetzins, and the court wanted them to give over information that had been given over confidentially.
The legal issue was whether or not Rebbetzins have the status of clergy and therefore are not obligated to give over information given in confidence. The other side was arguing that Rebbetzins can’t possibly be clergy since Orthodox Judaism doesn’t recognize female Rabbis.
The argument of the Rebbetzins and the Frum community was that women need to have someone they can confide in without having to worry about their confidentiality being broken, and often, there can be things that a woman might be more comfortable discussing with another woman, and that this is part of the role of a Rebbetzin.
In the end, they won the case. If anyone is interested, you can find articles about this case online by googling: Clergy confidentiality Rebbetzin. There was an article in the Yated about it.
My point here is that apparently clergy confidentiality does apply to Rabbis (and possibly even Rebbetzins).