I think the other answers' point about logistics is correct. But just to add another small insight:
There is actually at least one airline that charges by weight: Samoa Air. Samoa is also one of the most obese countries in the world. So its passengers which include both Samoans and non-Samoans (120 kg Samoans and 50 kg Japanese) exhibit a great variation in weight. With such great variation, it may be worthwhile taking on the additional logistical/administrative costs of price-discriminating by weight.
Like others have pointed out, there is actually price discrimination for shoe size, but done coarsely: Men, women, and children.
Within each group, they generally don't bother to price-discriminate more finely (i.e. by shoe size). That's because most men will fall within a narrow range of shoe sizes. Same for women and children. So the logistical/admin costs of price-discriminating by exact shoe size may not be worthwhile.
So perhaps in some country where there is very great variation in shoe size, it would also make sense to price-discriminate more finely.