Pets inbreeding?

No idea about inbreeding but the spay/neuter option is possible in the game as well in case you don't want them to breed.

The veterinary clinic can also be used to have your pet spayed or neutered the same way. Simply travel to the vet with your pet, sign at the kiosk (this time specifically stating what for), and then wait. That operation will cost a flat fee and cannot be reversed source

I'm pretty sure they won't inbreed.

They will inbreed sometimes. I've noticed that. But then again that's how they often breed "purebreds" in real life. Also, real animals don't usually care, when in heat. :D

They absolutely inbreed. I thought it was a bug but the bugs forum members responded that it is as in RL and intentionally designed in the game.