yes its possible to suddenly develop new allergies at any point in your life if your immune system isn't stable. they make shampoos and stuff to cut the dander your fuzzy friend gives off. try them and if your throat gets better its the cat. (if you can get puss used to the regular cleanings and be either diligent with the vacuum or get rid of your carpet you might be able to keep the cat even with a mild allergy.)
it could be allergies or maybe mono? if you have not had this before it can present as sore throat that is ongoing but is also generally along with fever, and tiredness and you had fever, a mono spot test if you have it would show up positive 2 weeks after symptoms first apparantly.