While mental illness cuts across all cultures, the symptoms are frequently culturally defined. For example, the Eskimos had something called ‘Arctic Fever’, where the afflicted tears off all their clothes and runs out across the ice as far as they can go. Needless to say, ‘Arctic Fever’ is not a phenomenon in places where it would be survivable.
Methods of suicide are also culturally defined. Yes, the statistics show that men are more likely to hang, shoot or gas themselves in cars, and women are more likely to use pills or slash their wrists. It used to be that people, more often women, would gas themselves in ovens. (A toxic component in cooking gas has since been removed.)
I suspect that the reason for the methods of suicide varying between the genders are simply because people are shown committing suicide these ways in fiction. Let’s face it, a filmmaker’s going to show their tragic heroine apparently asleep in bed, or in a bath of blood, while the macho dude’s usually going to go out in a macho way.