Dark City was a famous budgeting disaster because it spent way too much on sets and went rather over-budget, then did the terrible thing of not making that much money in the box office. The Matrix was able to spend so much of its budget on effects and effects innovation precisely because it got handed so many of Dark City's way too expensive sets trying to recoup some of the costs of Dark City.

They are definitely related, and I also think Dark City was the better version of Matrix 1 on a storytelling and aesthetics level, Matrix's innovative effects aside, but then I like all of the Matrix sequels a lot more than I like the first Matrix film.