Kamisama Dolls/WMG

Kuuko is Haruhi Suzumiya.
They both share the same enthusiasm for mysteries of the world, they both disregard authority when it comes to their club, and they both boss their club members around.

The Dolls are not really Gods. They're more like Empathic Weapons
Which change properties and special abilities for different contractors.
 * Confirmed, at least in part by episode 12. Which abilities are unlocked depend on the Seki.

The Dolls respond better to people with more guilt.
That's why Atsushi, who blamed all his problems on others, couldn't use it as well as Aki did, who felt remorse and sadness triple that of Atsushi. It's also why Aki became so strong, because he had been thriving on that remorse; and also why Kirio is stronger than Utao.
 * It could just be concentration. Atsushi lost control (and Aki was able to take control) because he panicked after killing Chihaya. Aki is strong because he doesn't lose his temper easily, while Utao is weaker than Kirio because he's able to push her buttons.

A kakashi's Leitmotif depends on its Seki.
When Kyohei was still Kukuri's Seki, it played a different tune than when Utao was controlling it.

Kukuri still responds to Kyohei.
When Kukuri attacked Takemikazuchi by shooting at it, it played the same tune as when Kyohei was controlling it during his flashback. Add that to how Kuramitsuha still responded to Aki when Atsushi tried to kill him with it, and there's precedent.
 * Appears to be confirmed.