Only, eh? Direct quote from the website of Nevada's Bunny Ranch: "Moreover, a Bunny Ranch encounter is entirely discreet, taking place in a private environment where the deepest, most intimate secrets of clientele remain confidential." It would very much hurt the Bunny Ranch's reputation if they reneged on that; it seems clear that confidentiality is something many of their customers value highly.
Just because the law doesn't mandate confidentiality doesn't mean the customers won't expect it and reliably get it.
My point was that "a service operates in the open" is compatible with "the customers of the service require confidentiality". Medical services are an example of that, an example in which people think that the need for confidentiality is so high that they made laws for it.
If they need a similar law for prostitution, they could pass one (I wouldn't be surprised if one already exists for a sufficiently broad category of "theraputic" services), but I doubt it would be necessary—at a brothel, only someone who verifies ages even needs to see a client's ID (and I suspect they could cover up the name); and as I said, it's strongly against the brothel's interest for there to even be rumors of non-confidentiality. I suspect a law would only move it from 99% privacy to 99.9% or something.
Just because the law doesn't mandate confidentiality doesn't mean the customers won't expect it and reliably get it.