According to an arcane New York legal statute, you're only allowed to call yourself a coop if your organization is actually a cooperative—that is, owned and operated by members. And if it's not? Says the law: "Any cooperative corporation may sue for an injunction against such prohibited use of the term." It's unclear whether the Food Coop would actually sue. Given that the manager raised the issue in a letter to the Brooklyn Paper, though, the grocery store clearly wants the borough to know that a lawsuit is an option.
The Barneys Co-op has been using the name for 25 years, plus they've had it trademarked, so they don't believe they're vulnerable to any kind of legal attack. But since only a food coop can bring a lawsuit against an organization abusing the term, it's possible that Barneys has been able to get away with it for the past quarter-century only because it hasn't antagonized any powerful hippie conglomerates. Now that Barneys is moving into prime Park Slope Food Coop territory, however, the battle is on.