I agree that not just guitars are called an axe, but sax's, trumpets, etc. Have heard this since I first started playing. Obviously a term that had been around for a while.
There is also a term for practicing your instrument that most players (especially jazz) have used for years - I know it dates back at least into the 1940's. it's called shedding. I gotta go shed... go practice. This came from early jazz days and the origin was that you couldn't practice in the house because it was too loud with too many distractions, so you'd go out behind the wood shed. It was known as woodshedding, shortened to shedding.
I haven't seen the tie in, but only makes sense that if you are going out to the woodshed, you take your axe.