Meowth views this Persian as a rival. Persian is the beloved pet of the boss of Team Rocket, and Meowth wants to kick Persian out and take its place as Giovanni's beloved pet, or the "Top Cat", as he calls it. Meowth's hatred of his evolved form was not started by this Persian, but his anger manifests against it more often than against any other Persian.
Giovanni also owns a Persian in one of the Pokémon games, Pokémon Yellow, which is highly based on the anime. He uses it mainly for battle, having been seen in all battles that the player has been challenged to. A Persian also appears in both rounds of Pokémon Stadium.
Giovanni's Persian seems intended to parallel other series in which the main villain owns a cat as a pet. For example, in Inspector Gadget, the villain Dr. Claw owns a cat named Mad Cat (it was named Sniffy in the 1999 movie). This trend seems to have started in the James Bond movies, where Ernst Stavro Blofeld is often shown to have a white Persian cat that he strokes in a similar fashion to Giovanni. His face also remains hidden during the first few movies, which Giovanni also parallels in the Indigo League saga until his battle with Gary.