Is a seed different from a nut?

There are two ways to define nuts, one botanical and the other culinary.

Botanical: It is a dry fruit containing one or two seeds, where the fruit does not open to release the seed. So, only indehiscent fruit are considered true nuts. eg. Walnut (image), pecans, chestnuts.

Culinary: A lot of seeds are called nuts even when they do not conform to the botanical definition. eg. Almonds, cashews(image).

Cocoa beans fall in neither category, so it is correct to say that the bag is nut free. Cocoa beans are harvested from a cocoa pod which is fruit like (image of cocoa pod with the seeds), so they definitely fall into the category of seeds and not nuts.

So the labelling on the bag is correct. As for the allergies, there may be people allergic to cocoa beans, but they wouldn't necessarily overlap with people who are allergic to nuts.

In the context of the labeling, they mean "tree nut" free, as many folks are allergic to tree nuts.

Technically, cocoa nibs are seeds, but they are not nuts. I have not heard of people being allergic to cocoa but that doesn't mean they don't exist.

Botanically, all nuts are seeds, but not all seeds are nuts.

Every fruit and many vegetables you eat are seed carriers. That's true from acorns to zucchini and there's a huge variation of types and structures. Many tree nuts seem to be descended from a common ancestor which contain a particular strain of protein that sets of allergies, while other fruits and vegetables (including cocoa) do not.

Nuts by definition are fruits with a hard shell, cocoa beans are not in a hard shell so they cannot be nuts. Not everything with a hard shell is a nut though, for example peanuts, which are not nuts at all.