This is something I have wondered for a long time. Growing up in a household of musicians everything was a pun. Dinner table conversation often included puns on names of pieces my father was playing with the BSO that week, or puns on names of soloists or conductors. Granted, communication with my family of origin was indeed odd, but didn't know it when I was a child. Since I didn't have a frame of reference, I thought every family communicated mostly in puns.
Puns were everywhere at Juilliard, and punning always happens whenever and wherever I play music with people. I love them, particularly when they just come out without any preparation. It makes any musical gathering feel like home.
I wonder if making puns is something that people do because they are musical (notice I didn't say "musicians," I said "musical")? My husband and kids are champion punsters, and they are also very musical. I wonder if the joy of the pun (which I just named "homophonophilia") has to do with a certain musical pleasure that comes into play in both the act of punning and the recognition of the pun.
Notice that I refrained from making puns while writing this post. You are welcome to make them in the comments, though.