Yes, a person can be allergic to rats and any other rodents. Usually the reaction is to the proteins in the urine or to the oils on the skin/fur. Antihistamines usually work well for hives/rashes.
Since rats are furry animals, you can be allergic to their fur. This allergy - like all pet allergies - can show in asthmatic symptoms, watering eyes, sneezing, and rashes.
It is not normal, unless you are allergic to rats. I lived with someone who had two pet rats, none of the four people who lived in the house ever had a reaction, however, when my boyfriend visited, within five hours he was covered in raised, itchy, red bumps.