
1 John 4:16 says, "God is love." Perhaps this is a good place to begin when trying to determine how we are like God. We have the capacity to love and be loved. Do animals have this feature? I am not a horse or dog whisperer, but I have grown up around animals. They do seem to have a capacity for caring and loyalty, but not love as we see expressed by humans. And every human has this capacity -- Christians and non-Christians, theists and atheists -- because we were all made in His likeness/image.
I have also seen other explanations as to how we are similar to God: self-awareness, the ability to think and reason. These are important explanations too. But they all have one thing in common in that they are more internal than external. They help us control our fleshly instinct to only care or worry about ourselves. I think this is a primary way we are made in God's image. We have the ability to put others first and to use this divine gift to set aside our own selfish desires (whether biological or psychological).
If we look at image and likeness in this light, we still see that we are unique when it comes to other animals. But we may share more with the animals of the world than we thought before. This might be a good thing as God commanded us to "rule over" the animal kingdom and "subdue" the Earth (Genesis 1). It gives us more of an incentive to take care of our planet (as if God's commandment wasn't enough) when we learn we are part of it.