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Brian’s jealousy and Peter’s desperation show a deep-seated human (and canine) need to be the "favorite," suggesting that our self-worth is often dangerously tied to how much attention we receive from others.
Ultimately, "Family Cat" suggests that "home" isn't just a place, but a delicate set of that can be toppled by the smallest—and fluffiest—of threats.
It highlights how quickly a person’s role in their own family can be undermined. Peter goes from the "head of the house" to an outsider simply because a more "charming" entity enters the space, proving that belonging is often conditional.
This specific episode, "Family Cat" (Season 19, Episode 19), is a sharp look at how a new presence can disrupt the established of a home.