Would a sore throat from rabies look and feel the same as a sore throat caused by a common cold? (postnasal drip, visible mucus in throat, etc.)

Yes: A sore throat is a general symptom of many conditions, If there is even a remote suspicion that you may have rabies you should seek immediate medical treatment. Sore throats are much more commonly seen as a symptom of viral sore throat, strept throat and allergic conditions

No: With rabies you would be extremely sick, frothing, crazy and about to die- horribly. With post nasal drip- you have a mild annoying tickle in the back of your throat.