Approximately 50,000 years ago, a meteorite crash-landed on Earth around an underpopulated region of Cape YorkW, GreenlandW. In 1978, a mineral excavation team uncovered an unknown virus within the rocks of the surrounding area and determined that it had been lying dormant inside the meteorite that hit Earth long ago. Despite extensive research, government doctors were unable to determine the alien virus' purpose outside the total elimination of human life. Akin to the theory of evolution, it was discovered that certain people possessed a quality that would enable them to survive when exposed to the virus, additionally granting them supernatural abilities. ~JJBA Wiki
Stands are the result of the extraterrestrial virus and the host's immunoresponse to it, which explains why Holly Kujo's stand was killing her and which makes it entirely possible that animals such as Iggy, Pet Shop, Forever, and Bug-Eaten in Part IV contracted the virus into their bloodstreams through various means.
On the surface, rogue stands such as Silver Chariot Requiem and Notorious B.I.G. in Part V, Stray Cat in Part IV and Anubis in part III might initially seem to present a glaring hole in this theory, but the fact of the matter is that viruses are microscopic life forms and the one in question is of alien origin, which means its interaction with both organic and inorganic matter after the incubation period is bound to have unpredictable results and allows for a wide range of creative freedom on Araki's part.
To sum up it up nicely, "It's an alien virus, I ain't gotta explain S#!t." - Hirohiko Araki