Animal sacrifice has been a common practice in cultures around the world. What we see in God's dealings with ancient Israel is directing that impetus of cultic sacrifice in a specific direction. God didn't get any pleasure from animal sacrifices, and the Old Testament rather plainly says as much. What God did care about was contrition, humility, and repentance. The point of sacrifice was the contrition behind it, not the actual sacrifice itself. Which is why God says, "I desire mercy, not sacrifice".
For Christians, we also believe that the sacrificial system served to point toward the fuller reality of Christ and His offering Himself in death. That the things of the Torah acted, as St. Paul says, like a tutor, pointing toward Jesus.
God doesn't have a problem with animals. They are His creatures, and He loves them. Jesus says God cares about every sparrow. And God's purposes in Christ is for not just human beings, but for all of creation.
And in Christianity this has, at least historically, been well understood. Animal abuse is, and always has been, a sin.
In one of the ancient legends surrounding St. Paul, he once encountered a lion in the wilderness, but rather than attack, the lion requested that he receive baptism to be a Christian. Paul obliged, baptizing the lion. Later on, while Paul is pit against wild beasts in Ephesus (an event he actually writes about in his letters) the same lion that he had baptized is forced into the arena. The lion and St. Paul then greet one another as brothers, and pray together.
Obviously, again, this is a legend and shouldn't be taken as anything more than that. But what the legend shows us is how early Christians saw themselves in relation to other creatures, even the beasts have a share in God's gift of salvation for the world. Even as the Prophet Isaiah when speaking of the Age to Come mentions that the wolf and the lamb shall lay down together, and the lion shall eat straw like an ox, for on that day peace has come to all creation.
In many churches the Feast Day of St. Francis of Assisi is celebrated by the
Blessing of the Animals. Which means, yes, people bring their cats, dogs, hamsters, lizards, and other furry, feathery, scaly family members to church and receive a blessing.
-CryptoLutheran