"Countries with legalized prostitution are associated with higher human trafficking inflows than countries where prostitution is prohibited. The scale effect of legalizing prostitution, i.e. expansion of the market, outweighs the substitution effect, where legal sex workers are favored over illegal workers. On average, countries with legalized prostitution report a greater incidence of human trafficking inflows."
Whether you hide your patronage from people who know you is a different matter. (The medical system is a good example in this respect: hospitals and other facilities are absolutely not hiding their presence from anyone—quite the opposite, really—but the fact that you visited Doctor Jones to treat your potentially embarrassing medical condition is considered very confidential information.)
Just because the law doesn't mandate confidentiality doesn't mean the customers won't expect it and reliably get 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.