God has entrusted animals to the care of man (Gen. 2:19-20). St. Francis is a great example of how we can love animals, without overdoing it. We sometimes can have a great bond with an animal.
There are major differences between humans and animals, such as the fact that animals follow their instincts while humans can think rationally. The Bible refers to animals as species, but refers to people as unique human beings. God deliberately created each one of us. Animals have no purpose outside their earthly life. In heaven we will be completely happy because we will worship God forever. Therefore, we will not miss our pets there.