The rationale is "it may make you feel better 🤷‍♀️". There is no evidence to indicate it's therapeutic, but if it causes you to drink more fluids, and if you subjectively feel better even if you aren't necessarily getting better, then there's no harm in recommending it.

There is actually no product on the market, prescription or over-the-counter that is known to meaningfully reduce your cough symptoms or the time it takes for you to recover from a cold. This includes things like Zicam and OTC cough syrups (dextromethorphan, etc.). The more these drugs are studied the less confidence there is that they a single thing to help you. A lot of these remedies came into use before we have today's standards for establishing their safety and efficacy.

The placebo effect is a very real phenomenon. It's not just "you're imagining that you're better". The belief that you are being helped can result in very real, measurable, physiological improvements. This typically happens because of a reduction in stress associated with the illness. When you anticipate more (and worsening) symptoms, your apprehension can release stress hormones that exacerbate them. The certainty that taking a drug will make you feel better can reduce or eliminate that stress, giving you better outcomes.