I would certainly consider them such; I don't find them in my grocery store, though, so if you can name some, please add them. I deliberately did not include seaweed, though, such as Japanese Nori and Hijiki, because I think English speakers generally apply the term "seafood" only to animals. Algae may also be a stretch for some.
Actually, i used the word algae because i forgot the term seaweed.
Well, there's a Welsh dish from seaweed, known as 'Laver bread' IIRC.
....and sodium alginate is a very common food ingredient ,as a thickener and gelling agent etc. It's not regarded as a food item in its own right, though