Khajiit are not enslaved in Elsweyr as far as I know - but your cat could maybe have been a slave in Morrowind - although slavery was outlawed even there by the time of Oblivion.
In the Deserts of Anequina I seem to find plenty of Khajiit slaves, I found a slave pen in Orcrest where they sold slaves using an argument like a ''they are convicted prisoners, having to pay for their sins'' sort of thing.
Yes, well, DoA's honestly not very lore accurate.
Slavery may possibly exist in Elsweyr but it's certainly not legal as it was in Morrowind.
There's some support in TIC that slavery exists in Elsweyr, but not to any large extent, Khajiit are largely against slavery because of their treatment by the Dunmer (and to a lesser extent Altmer).