What feature of rabies pathophysiology causes hydrophobia? Why is hydrophobia unique to this one particular type of viral infection?
Rabies causes hydrophobia in the encephalitic stage which means when it affects the brain and causes swelling and inflammation of multiple areas of the brain. Hence, it affects the complex areas in the brain needed for swallowing. Initially into the course of the disease, the patient has involuntary contractions of neck muscles when he drinks water. At later stages of disease, the patient starts contracting his muscles even at the thought of water. Rabies virus is a neurotropic virus which means that it travels through nerves because of its preference to attach to acetylcholine receptors in the neurons. That is how the virus spreads from the bite site to the brain(via the nerves).
No doubt, the selected answer is sufficient for the question. But for the ones more interested in citations, here is a more credible answer.
When the rabies virus attacks someone, it begins with muscle cells, so that it is not detected by the host's immune system. There, it binds to neurons at neuromuscular junction1. Rabies virus, being neurotropic, binds preferentially to neurons, specifically the acetylcholine receptors on neurons. After binding to neuromuscular junction, it uses retrograde transport to travel to the axon, as its P protein interacts with dynein in neurons. As soon as it reaches in neuron cell body, it rapidly reaches the central nervous system, where it replicates in motor neurons and quickly reaches the brain2. From here, it travels to peripheral and autonomic nervous system and then reaches the salivary glands3.
Now, lets come from the how part to the why part. Rabies virus gets accumulated in salivary glands of the host so that it can get transmitted further on to the next host. And, as one might expect, drinking water or other fluids can decrease the virus's ability to get transmitted4. To prevent this, the virus causes painful spasms in throat and larynx so that not only saliva production in host is greatly increased, but also drinking, or even thinking about drinking, causes excruciatingly painful spasms in throat. Since the muscular movements during drinking (or eating) are involuntary, and the virus has already infected autonomic nervous system and motor neurons, it is thus able to control involuntary muscle movements of throat and larynx in the host5.
Gluska, Shani & Zahavi, Eitan Erez & Chein, Michael & Gradus, Tal & Bauer, Anja & Finke, Stefan & Perlson, Eran (August 28, 2014). "Rabies Virus Hijacks and Accelerates the p75NTR Retrograde Axonal Transport Machinery". PLOS Pathogens. doi:10.1371/journal.ppat.1004348.