No and Yes. Sea water is essentially, truly just salt and water. Other things like magnesium, etc are there too. But true sea water is salt water. However, just about anything can be found in sea water, so if someone is having a problem with a reaction to it, then its likely to be some contaminant in that local source. Something dumped or leached into the system is the likely culprit.
No. A person is not allergic to sea water, but may get itching and / or rashes from things that are in the sea water. Sometimes, a person can get hives (an allergic itchy rash) when he swims (or showers) in cold water, due to the cold temperature. A person with very dry, sensitive skin might have tiny dry cracks in his skin that allow salty sea water to seep into and cause itching or irritation.