The writings of Martin Luther contain a lot of scatological references. Was this common at the time (i.e., part of wider culture), or a personal quirk of Martin Luther's?

I'm not an expert on the continental reformation, but I do know that there were others who used scatological humor and discourse in their writings.

My absolute favorite example is a quote from Sir Thomas More (of Utopia fame), in his Responsio ad Lutherum. From Volume 5 of the Complete Works of Thomas More, here is the translation:

...others will be permitted, on behalf of his English Majesty, to throw back into [Luther's] paternity's shitty mouth, truly the shit-pool of all shit, all the muck and shit which your damnable rottenness has vomited up, and to empty out all the sewers and privies onto your crown...

Really, the entirety of the Responsio is a glory to behold, in that it is filled with venom, hatred, and insults towards Luther, while trying to also argue theology.

Chapter 4 of Matheson's Rhetoric of the Reformation is a discussion of published dialogues during the early reformation, and he comments on the use of shitty language. Many of the dialogues are written in the vernacular, and even mock the Latin language as a language of obfuscation (p. 103). Some other examples of scatological terms are, when Heinrich von Kettenbach calls Thomas Aquinas, Thomas Shit-Wine and Hans Bechler talks about shitting in a bishops mouth (pp. 103-106).

Looking later in the century, there is a great study in Cartwright's Companion to Tudor Literature on Gammer Gurton's Needle (1551-1552). The work was a comedy play, and references shit frequently - the characters are known to shit their pants as part of the humor.

All of this is meant to indicate that it was not a personal quirk of Luther's to include references to scat in their writings. It occurs in the distinguished Latin of Thomas More, and in the vernacular dialogues and plays. A major reason we see it cropping up in the sixteenth century is because of the press of evangelicals (Protestants before the term had wide use) or Lutherans to connect with ordinary people. English and German were the language of the common people, and authors wrote in a way that the common man at his plow would understand - and at times that meant scatological references or humor. William Tyndale, referencing Erasmus, once said "I will cause a boy that driveth the plough shall know more of the scripture than [Catholic devine]", (Quoted in Bridgen, New Worlds, Lost Worlds, p. 131).