In a weak moment, I was about to say 'Yes, wordplay is punny'*, but then I started wondering whether there's a word that distinguishes 'true' puns (i.e. those that use pure homonyms) from those that rely on a similarity of pronunciation, or maybe just even rhyming for their effect.
For example, there are the endless fish-based jokes that talk about 'the sole plaice I <extemporise on flounder, roe, fry, skate, pike etc>, but that's not the same as e.g. 'In Cod we trust'.
*OK, I said it anyway. Sorry.
This is a bit profound, sorry I didn’t respond sooner. Well I don’t know of such a word, but maybe it exists, or we could invent it. <scratches head>
One of my top favourites is “Many are cold, but few are frozen”. I did make that up, but as you may guess an internet search confirms many others beat me to it.
There is of course “The piece of cod, which passeth all understanding.”