Answer: You are right. Envy on the level of our world is an ordinary egoistic feeling, a competitive means of struggling for a source of pleasure, whether it is an animal receiving pleasure from its owner or a person receiving pleasure from a beloved.
The wisdom of Kabbalah speaks about the concept of envy, and especially the concept of shame, in regard to emotions that a person feels when he attains the attribute of love and bestowal unto others. This means that he attains the attribute of the Creator (nature), meaning the human level, Adam.
Unless we reach this level, we are considered animals, just like your cat and dog. The nature of our feelings is the same and it stems from our egoistic fears of remaining without a source of pleasure (food, sex, etc.).