I guess folks don't use this phrase that much these days. But I've heard it all my life. It comes from the Punch and Judy puppet shows which have been popular in Britain since the 1600s.
I'm sure you've seen one of these at some point. The big nosed, hunchbacked Punch is married to Judy. There's always a lot of yelling and screaming. Before they got cleaned up for the more modest audiences of today, the stories actually involved multiple murders and infanticide. It was sort of like Lifetime Television for Women on the small stage.
Punch is always concocting evil schemes, and when they actually work, he struts around the stage, all full of himself. So, to be pleased as Punch or "proud as Punch" has meant self-satisfied for at least a couple of hundred years.