In 'The Cat Returns', the gentleman cat is Baron. Baron is a refined, anthropomorphic cat who helps Haru escape the Cat Kingdom after she is unwillingly drawn into its royal affairs. He is courteous, brave, impeccably dressed, and skilled with a rapier, giving him the air of a traditional English gentleman.
The character originated in Aoi Hiiragi's manga Baron: The Cat Returns and was also introduced in Studio Ghibli's earlier film Whisper of the Heart. In that film, Baron is a figurine in an antique shop and becomes part of Shizuku's imaginative storytelling. The Cat Returns expands him into a living character and makes him the leader of the Cat Bureau.
A common mix-up is confusing Baron's title-like name with aristocratic alternatives such as Earl, Duke, or Prince. The Cat Kingdom has a king and prince, but the gentleman cat himself is specifically Baron. His full name is Baron Humbert von Gikkingen.