One of the issues with "Sockeyes" is that it's an ugly fish. Even by fish standards, it's really ugly. The word itself also sounds somewhat grotesque.
Seattle Seals checks the alliteration box that I personally enjoy (Pittsburgh Penguins, Philadelphia Flyers, etc.), seals are a cute animal that most people like (similar to penguins), has great mascot and merchandising potential, and possibility of a cool Penguins-eque logo and jersey, etc.
At least the Marlins (Miami, MLB) are named after a cool looking fish. But even then, nobody likes that team or buys their merchandise. Do you really want to wear a jersey with a fish on it? Especially a really ugly one like a sockeye?