No. realistically, humans would use pig intestines as protection. Elves arent as fertile as one would think and Argonians need to lick a hist tree or something before mating to have any young.
Khajiit..well i dunno barbs would certainly be a turn off for some females
Doesn't seem to be a turnoff for female cats, now does it?
I'd say they likely use pig intestine, sheep skin, etc. for condems like in the olden days, or medicinal herbs that work like the pile and affect hormone levels. One could surmise, considering where the pharmacists get most of their chemical compounds for their pills. Lots of it comes from animals and plants.
It could also be that growth rates of young in Tamriel are high. But this is unlikely.
And the elves are, truly, damn near infertile. Most elven women are lucky to have two children, if not one. However, oddly, it is mentioned in
Barenziah that having frequent intercourse with a human will make an elf of either six more fertile. Which explains Barenziah's frequent pregnancies.
The Elves also seem to not suffer from menopause, or experience it MUCH later in life than humans. We KNOW for a fact that Khajiits birth young, because it is explicitly stated that a the lunar phases during the Khajiit's birth determine it's breed. So obviously, they must be born, and thus have children.
It could simply be that the Gods hide the children because of all the violence in the world?
And magic is plausible, as well. But likely only used by the more masterful mages, and not given to the public as it would be too risky.