Simple: Control their breeding.

The difference in natural social instincts may be a factor, but by far the reason cats are only partially domesticated is that even though they have been domesticated to some degree, for the vast majority of human history, we let cats figure out this part all on their own. Apparently at the first cat show in 1871 there were only five distinct breeds of cats. It's really only very recently, in the age of the modern housepet, that we've really taken control of cats' genetic destiny — and even then very loosely, as we many domestic cats are allowed to breed with their feral neighbors.

By contrast, humans started selectively breeding dogs for desired traits literally before history started.

So that's pretty much the difference. See Why cats never became man’s best friend for more on this. If we want a different history, simply remove this option — keep your domesticated cats under closer control and breed for the desired social and human-friendly traits.

Simple: Control their breeding.

The difference in natural social instincts may be a factor, but by far the reason cats are only partially domesticated is that even though they deign to live with us, for the vast majority of human history, we let cats figure out the "where do babies come from?" part all on their own. 

Apparently at the first cat show in 1871 there were only five distinct breeds of cats. It's really only very recently, in the age of the modern housepet, that we've really taken control of cats' genetic destiny — and even then very loosely, as many domestic cats are allowed to breed with their feral neighbors.

By contrast, humans started selectively breeding dogs for desired traits literally before history started.

So that's pretty much the difference. See Why cats never became man’s best friend for more on this. If we want a different history, simply create a line of domesticated cats which you don't give this option — keep them under closer control and breed for the desired social and human-friendly traits.