It’s the same problem as with XML. Tiled does not keep the JSON document in-memory but rather has its own data structures, so there is no place to store arbitrary JSON data apart from the custom properties.
Even if Tiled would keep the original XML or JSON documents in-memory, I’m not sure how it could keep unknown sections intact in general. If you have some solution in mind for this you could always try patching it in. It’s probably easier with JSON than with XML, but either way I think you’d have to litter all existing data structures with custom JSON / XML data holders or handles.