Turns out it has to do with osteoporosis in older subjects.

Measuring bodyfat in a water tank requires weighing a person outside the tank, and immersing them in water and weighing them again. The densities of bone and muscle are higher than water, and fat is less dense than water. In other words, fat floats. So supposedly the difference in weight is accounted for as bodyfat.

But this method assumes that densities of nonfat mass, like bones, are constant. However, athletes tend to have denser bones and muscles than non-athletes, which may lead to an underestimation of body fat percentage. On the other hand the body fat of elderly patients suffering from osteoporosis may be overestimated.

So I guess this explains why the Omron device I have at home wants to know your age -- it must make some kind of adjustment when it calculates bodyfat percentage relative to bone mass.

Still doesn't completely explain why you'd input gender, unless they assume a higher rate of osteoporosis in females. But that's just a guess.