Does cheese have to be made with rennet from the animal whose milk it is?
No, the article is bogus. There are many ways to coagulate proteins (although the exact method chosen will make a big difference in what the final result looks like). Heat, acid and enzymes all work for coagulation. And while you can't use heat to make milk, both acid and enzymes from various sources is OK. The enzymes suited for cheese making are called rennet. 

There is some difference between proteins in the milk of different species, so not every rennet will work with every milk. But the fit is nowhere as specific as the article you linked implies. Bovine rennet works for sheep's milk too, I have personally helped in the process of making that cheese. There is, as you mentioned, rennet from non-animal sources. I hadn't heard of plant-derived one (although I'm not surprised), but bacteria-derived rennet is widespread in the industry. 

I also remember reading somewhere about actual human cheese, although I must admit I don't remember it being a reliable source, and have no first hand experience with it, so I am not 100% confident about it. Still, from a chemical point of view, it should be possible to turn human milk into a solid slab No guarantee about tastiness, and you may need to try a bit around until you have found a suitable coagulation agent, but it should be possible to create one without chopping up baby stomaches. 

The argument from the article wouldn't hold anyway, even if the species' rennet was required, because there is a wide variety of acid coagulated cheeses which does not use rennet at all. 

Seeing that the article you linked is presented as an answer to a two-year old's question, maybe just the person who answered thought that they needn't bother fact checking for such a "simple" task. On the other hand, the strange statement, coupled with an article ostentatiously aimed at a two-year-old but mentioning protein coagulation, make me think that it may be some kind of joke rather than intended as a serious description of the problem.