When guitar players do this around their band members, they are basically talking in slang. This is a type of language that can be loosely defined as a mode of communicating that is exclusive within a particular group of people that have a common understanding of the meaning of the words. In this example, the band members immediately understand that when the guitar player says axe, he means his guitar. Slang can be found in a lot more places and can be as creative as the group wants it too.