As far as we know the simile "as pleased as Punch" and the variant "as proud as Punch" does refer to Punch of the Punch and Judy Show. His catchphrase is the self-satisfied "That's the way to do it!!" and he chortles with glee. "Huzzah! Huzzah! I've killed the Devil!" shrieks Punch in his final triumph.
The following is extracted from Brewsters Dictionary of Phrase and Fable 1898
1. "You could skip over to Europe whenever you liked; mamma would be pleased as Punch." R. Grant
2. ... is kicked out; (3) Death is beaten to death; and (4) the Devil himself is outwitted. 4. Pleased as Punch
Pleased as Punch Meaning : Very pleased.
Origin: From the Punch and Judy slapstick puppet character. Punch originated from Pulcinello an Italian puppet with similar characteristics. In Punch and Judy performances the Punch character is depicted as self-satisfied and pleased with his evil deeds.
ref: Brewer's Dictionary of Phrase and Fable, 15th edition.