My question is why does it have to be a plane?
When people talk about the focal plane in a photographic context, they usually mean the image plane, a.k.a. sensor plane, the area where the image is formed. Since film and sensors are (usually) flat, a field of focus that is also flat is desirable. The field of focus is often not flat, which is one reason that many lenses are sharp in the center but softer at the edges, but lenses often include elements that flatten the image as much as possible.
I'm guessing, though, that you're talking about the focal plane in front of the lens. You're right that the space in which the subject is in focus is shaped like a sphere. Wider lenses see a larger portion of that sphere, but they usually also have greater depth of field. Long lenses see only a small part of that sphere, which means that the field of focus is close to flat.