Is it possible to have a sudden shellfish sensitivity without it being an allergy?

Sure, it's possible: Many things are possible in life. Each person's food "sensitivity" should be evaluated to get to the root of the issue. That's because "sensitivity" means different things to different people. For example, various people might have different symptoms such as: unusual taste, sensation, or tingling in the mouth or throat, nausea after eating, tummy ache, vomiting, itching on the skin, rashes, etc.