The point is 々 does not mean simple plural. It suggests there are a lot. It'd be more like collective nouns, if I have to compare to English. 家と家の間 : between two or several houses. 家々の間 : between many houses (probably making turns randomly, so the dog's path will be not straight. That's why Kunio-san uses 走り回った instead of 走った。) Also, this was not part of your question, but you can use で only when the dog runs around in some limited area, such as parking lot, cul-de-sac, backyard, or between houses of some area, at most. If the dog runs though and goes away, you have to use を。I'm not sure which one you meant to say.

「犬は家々の間で走った。」「犬は家の間で走った。」ともに普段使わない表現です。 ・"間で" means "at/on the place between A and B" ・"run" includes "pass through".　 １．In case of "the dog ran around" If the space of specific two houses is wide, 犬は家の間で走り回った。 If the space is narrow, the dog runs among houses. 犬は家々の間で走り回った。 ２．In case of "the dog passed through" If writer's interest is narrow area of houses, 犬は家の間を走り去った。 If writer's interest is wide area among houses, 犬は家々の間を走り去った。 - end -