The relationship between forest and wildlife is based on the same principles as the relationship between any animal and the ecosystem in which it exist. There is an intrinsic relationship between organisms and their habitats, which is one reason environmentalists are deeply concerned about the destruction of habitats, such as deforestation, because one result is the harm done to the wildlife in that habitat.
The wildlife whose habitat in the forest are those who find the food they are adapted to, and the areas for rest, nesting, and the other elements that support life. It is the case that some wildlife can be brought to urban areas and adapt to it, but it is also true that the attempt to domesticate many wildlife species does not result in domesticated animals. This also helps explain why domesticated animals, such as cats and dogs, were derived from wildlife, but it took many reproductive cycles of careful breeding as well as introduction to urban-type environments to end up with a domesticated animal.