Until 1974, Wolverhampton and Stoke were both in Staffordshire, so it was an intra county rivalry, one in the south, one in the north, and people in the middle were split (including the town of Stafford). That's why the rivalry is probably a generational thing.
In 1974, a new county was created to incorporate many large towns and cities in the West Midlands region, including Birmingham (historically in Warwickshire). Confusingly, they called the county "West Midlands", which is also the name of the region that it and the surrounding counties sit within. Wolverhampton is in the North West of the "new" county, meaning the focus became inwards (east and south, towards Birmingham) rather than outwards towards Stoke and Port Vale.