Modifier prefixes are added in front of the "The" in its name, which can result in a grammatically incorrect name such as "Keen The Axe". It shares this trait with all other items that can obtain modifiers and whose names begin with "The". This error is not present in the Old-gen console version.
The Axe was added as a tribute to Dimebag Darrell, guitarist and member of the American heavy metal band Pantera, which Cenx has stated is her favorite band. Fittingly, Plantera, the enemy referencing Pantera, drops it.
Its name and sprite refer to electric guitars being nicknamed "axes" in rock and heavy metal music culture. Its dual role as a hammer comes from how some classic "hard rockers" would famously smash their guitars on stage as part of their acts.
Before becoming rock slang for the guitar in 1967, the term "axe" was first used as jazz slang in 1955 (possibly earlier) for the saxophone, and later the trumpet (for its history; see here, and here).