'Bor han i miT område?' would be a more commonly heard, but 'region' isn't wrong. Although there is no 'size-definition', most people would think of an area (område) as being smaller than a region. If you want to know if the man lives inside a quite large part of the country, the word 'landsdel', which means part of the country, would be the most natural word. Besides, if you ask 'Bor han i min region?' a Dane would most likely think of the 5 local authorities in Denmark, who runs the health service systems in the geographical areas each one of them is responsible for. So implicitly the question could be interpreted as a question of which health service system he 'belongs' to.