And that song was from 1969 - in a phase when the Beatles 'were exploring folk song/tales/ballads'

[AI] The song is about a student named Maxwell Edison who commits murders with a hammer, with the dark lyrics disguised by an upbeat sound. McCartney described the song as symbolic of the downfalls of life, being "my analogy for when something goes wrong out of the blue, as it so often does".

So, some sidebar bits:

Why does Paul McCartney laugh in the song “Maxwell’s Silver Hammer”? [Quora] Originally Answered: Why does Paul McCartney laugh in the song ‘Maxwells Silver Hammer’? “The story goes like this: John didn’t like the song, thought it was one of Paul’s “granny songs”, so he left the studio when they were recording it. As Paul was singing it, John stepped back in and mooned Paul. That’s why Paul laughed.”

It sounds like evil laugh inserted.

Anyhoo:

Seeing as himself adopted the name to hide his vile and possibly already murderous identity from SSSR (Soviet) authorities, it would be a 'fitting psuedonym'. Or, maybe the Beatles knew something (it wouldn't surprise me, given their meteoric rise to fame, and the mad scene presenting itself as everyone wanted to know them (inluding fresh spooks), and party with them (see B-grade movie "Help") - or maybe the elite-spooks recruiting or farming young 'Maxwell', chose the name, because of the macabre black humor of the song (similar to Mack the knife) that handily describes the narcissistic, sociopathic character of a this uniquely qualified 'defector'.

Weirdly, Maxwell made his fortune printing Nazi science that had not been published due to certain interruptions in their timeline. Maybe that's why there were mysterious murders involved? Or maybe the song was just a silly attempt at singing in a jolly fashion about mass-murder IDK.

Folk songs are like that, they have a dark side, that makes people remember.