What is the name of this manga about a dog who gets his tongue cut off and his owner is a teen whore?

That one is called Deep Love: Pao no Monogatari.

There are a few stories that continue from a different perspective, like Deep Love: Ayu no Monogatari (from Ayu, the owner's viewpoint), Deep Love: Host (from Yoshiyuki, the boy she falls in love with's point of view), and Deep Love Reina no Unmei (from her friend's point of view).

All of these stories are extremely touching and make me cry everytime I read them. It really is a pity they haven't been officially translated into English, but you can find scanlations online. They give a realistic depiction of animal cruelty and how some people are put thought extremely difficult situations.