Inugami was a dog before he was an inugami. He was abandoned on the street wherein he eventually died and became a ghost. He's obsessed with Kohina because he believed that Kohina was the one who gave him an umbrella when he was a dog while muttering "Dolls are not bothered by rain.".
In the manga, it was later revealed when Kokkuri-san was "time travelling" that Kokkuri-san was the one that gave Inugami the umbrella, not Kohina which shocks Kokkuri-san and makes him feel guilty for ruining two people's lives.