I'm sure that's not what happened. The populous, if I remember correctly, was spastic, so they might have made exceptions. However, I can't say anything for sure. For all we know, they were all called "The unnamed ones" or something like that.
We don't actually know what happened to any Khajiit born at that time, if there were any. And if the Thalmor know, they're not telling. We have depressingly little infomation available on the Void Nights.
- The referernce to Elsweyr having it worse during this time was probably nothing more than historians acknowledging that the moons have an important role, not hinting at what MAY have happened during that time.
- This thought makes the above pointless anyways, but everyone wonders how the moons disappeared, and how they reappeared. My thought is that something (not the Thalmor) just hid them from sight (yes, from everyone). There is, as far as I know, no idication things on Nirn were impacted by their disappearence (like there should have been, two giant bodies just vanished, on Earth, this would cause serious issues and we'd all be dead in a week). So some strong magic was used, most likey Daedra or Aedra in origin, what else could cause such an event? Not simple mer or beast magics.
And IF the Thalmor were responsible, don't you think they would gloat a little more about it instead of waiting until they reappeared?
Also, in response to the actual question, what happened to the Khajiit? They were born normally. Any Khajiit would have been born as if the moons were there, because they were, we just couldn't see them. Like air, we know it exists, just can't see it.
I agree that the Thalmor are unlikely to be responsible, but I believe Mer or Beastfolk may actually be able to do something like this, take for example the Psijiic order, they knew about the Eye of Magnus, and once it became unstable, they took it to safeguard it. Just like that, an artifact possibly created by Magnus, the God of Magic. Now i don't know how they can safeguard such a powerful artifact without being ridiculously powerful themselves.
Masser and Secunda both are parts of Lorkhan. a dead god. What could possibly cause him to Disappear or Vanish from sight?
^ They are possibly parts of Lorkhan's corpse, according to the Lunar Lorkhan. However, the Khajiit don't see it that way. Jone and Jode are the moons, and are separate from Lorkhaj.
However, it is worth noting that Lorkhan is always the Missing God, described as "the upstart who vanishes" in the Commentaries on the Mysterium Xarxes. Although I very much doubt that it has anything to do with the moons' disappearance.
It was a lways my assumption that the Thalmor hid the moons with the intent of getting the Khajit to join the Altmer Dominion. the Dominion needed more troops, and with the Thalmer claiming to have "brought the moon back" the khajit practically worshipped them for doing so. The Thalmer hid the moons with the sole intent of being credited so the Khajit would join there forces. My only question here is why would they wait two years????
Why did the Thalmor wait for so long to "bring back Jode and Jone"? This one finds the answer simple. The Thalmor waited till the fractures of all fears grew wide enough to assure the shattering when their hammer fell so they could remake us to their image of a faith servant..A slave but of a prettier name. Merrunz rend them all.