I suspect the short answer is sloppiness. The supermarket is Giant, but the sports team is the Giants. People hear "The Giants" all the time, and then, sloppily, pass the plural "s" on to the name of the supermarket.
Similarly, many places have "Springs" plural in their names: Colorado Springs, even Silver Springs, Florida. Silver Spring (MD) (no "s") is kinda rare. And people are kinda sloppy. So people say "Silver Springs" and get the name of the town almost right.