It doesn't even have to be an allergic reaction - idiopathic urticaria (hives just because) is a real thing, and can be harder to treat than hives due to an allergy.
And it may not go far enough to be neurotic, but every time I'm exposed to somebody with some kind of infestation (scabies, head lice, body lice...), I itch like crazy. I don't have to actually catch it, just be around it, and knowing that it's psychosomatic doesn't make it itch any less.