Yeah, but people don't breathe through our skin in any meaning of the word. If you paint yourself or put on a skin-tight body suit, the result is the same: you have reduced the ability for heat to transfer from your body to the surrounding environment and may have inhibited the production of sweat. So, your body starts reporting that it is stuffy and it can't "breathe", but it does not do this when you are submerged underwater.
As for how a mammal could respire anaerobically, I have a few theories on what they mean:
1) Monotremes, like platypi, lay eggs, and although eggshells have pores that allow oxygen transfer, I don't imagine it is sufficient for normal mammalian fetal growth (Although Wikipedia says, "However, the egg is retained for some time within the mother, who actively provides the egg with nutrients"). Remember, being warm-blooded means you consume more food, energy, and oxygen than cold-blooded creatures, and a platypus fetus would be growing rapidly and consuming its food reserves equally fast (although Wikipedia says, "[A monotreme's] metabolic rate is remarkably low by mammalian standards, although the extent...is uncertain."). Perhaps they have a prenatal method of consuming this energy without the need for so much oxygen, or a partial anaerobic system. I don't know, I'm not a zoologist.
(Interesting side note here, I read this on Wikipedia:
Wikipedia wrote:The key physiological difference between monotremes and other mammals is the one that gave them their name; Monotreme means 'single opening' in Greek, and comes from the fact that their urinary, defecatory, and reproductive systems all open into a single duct, the cloaca.
2) Farts. Our bodies (or more specifically, the bacteria in our bodies, which kind of debunks this theory) produce gases, including methane, without the need for oxygen. Maybe that's their reasoning.