I don't think either is going for each others market - well not in the freight business anyway. Both companys are expanding - the market can support a lot of growth as there doesn't seem to be any shortage of freight to fly from Asia. Direct freight flights from Asia to Europe don't make sense. You would have to carry so much fuel that the load would be too low. Tech stops in India or the Middle East will be the order of the day whether a Classic or a 744.
Cathay and Dragon aren't owned by the same company. Cathay is a publicly listed company with various shareholders. Dragon have various shareholders (including Cathay) but are largely owned by the Chinese government.