No, Nightmare and Dream were both made by the same person, and have a canon storyline by their creator explaining their backstory together. They are from the same AU. They were guardians of this sacred tree, as well as brothers, and Nightmare accidentally corrupted it, and then himself when he ate the corrupted apples- Which is why he's a jerkface. Ink and Error are just enemies from the beginning, which Ink being a creator and protector and Error being a destroyer.