I don't envy you this very difficult situation. Perhaps the reason that the neighbors have a history of this type of behavior is because no one ever confronts the behavior and simply says "NO!". You should not be expected to make the best of a situation for which you are not responsible. It is not worth your mental health to spend a trip trying to avoid or accomodate people that you didn't invite.
With that said, there are honest, respectful ways to say, NO!! I would ask your friends to explain to their neighbors that there has been a simple miscommunication. If your friends confront the "unfortunate situation" instead of the neighbors, perhaps the fallout will be less. If your friends believe that this will pose a great problem for them in the neighborhood, then ask if they mind if you confront the neighbors. Explain that the cruise is an annual trip for your two families to re-connect and it is planned that way for just your two families to share old memories and inside jokes and create new memories together. Apologize if the neighbors misunderstood but your plans are firm.
Also, (rather than blame the neighbors for being so pushy) apologize that
DCL allowed the neighbors to link to your reservation without your consent. Obviously, if the friends had known up front what the neighbors plans were, they could have corrected the misunderstanding. That is a nice way of saying "you should have asked!"
If the neighbors still choose to cruise, it is their choice but you have no requirement to include them in any activities because you told them the truth. Then, I would call DCL, verify that DCL linked the ressies. If in fact, they allowed this to happen, how are they going to correct the situation. I think linking reservations is a great service but only if all parties agree . De-link the seating and ask to change stateroom assignments and if there are no more rooms in your category that meet your needs: request an upgrade due to an untenable situation that DCL helped create.
I wish you the best and I hope your vacation turns out to be simply magical!!