It is the same gene (MCR1 melanocortin receptor) but there are multiple independent mutations that have caused red hair. The reason? Mutations happen all the time. Being red-headed is not a disadvantage if you're not facing a lot of sunlight (redheads don't tan well). As a result, scientists speculate it arose and did not die out in Northern latitudes where it eventually became a well recognized trait which is preferred by some and disliked by others. Meanwhile, it probably arose multiple times closer to the equator and was selected out into a very rare trait.

There are actually mutations in different genes that can cause red hair, such as one in Papa New Guinea. But the vast majority of people with red hair have 2 mutations in MCR1.