Interesting question! Meat is defined by Webster dictionary as "the flesh of animals, particularly mammals, as opposed to fowl and fish" (http://www.merriam-webster.com/dictionary/meat). Based on that, the answer to your question would be no, seafood does not constitute meat.
Generally speaking though, when people say that they do not eat 'meat', I have noticed that many imply they are vegetarian, i.e. do not eat any flesh, fowl, fish or mammals.