There were a lot of different typefaces available. All manufacturers had their own typefaces and many manufacturers offered typewriters in multiple typefaces.
By the 1960s companies like IBM had figured methods to make changeable typefaces and Typewriters like the Selecteic and Selectric II had not only easy mechanisms for changing out the type elements allowing a switch between something like Courier to Letter Gothic. And they had typefaces like Symbol to allow people to type using mathematic symbols. By the time of the Selectric II, they even made it possible to change typefaces with the switch of a lever, rather than physically swapping type elements. So an Israeli typist, for example, could switch back and forth between a Latin language and Hebrew without any disruption. Or an editor could switch from a 12 character per inch typeface to a 10 character per inch typeface for emphasis, much like bold or italics in modern word processing. I hate to link a blog, but you can see a sample of some of the typefaces available on the original Selectric here.
That said most typewriters do have a certain look to them. When typewriters first came out they had a variety of typefaces. The Remington No 1 used an all caps san serif like Ariel. But the slab serif designs were very popular in the 19th century. So many typewriters went with that. One characteristic that was especially common early on was monospacing. By giving all letter the same spacing so that “i” and “M” required the same amount of space on the page, it simplified the mechanics. It’s far easier design wise for each character to advance the page an exact amount, usually 1/10th or 1/12th an inch. So most typewriters typefaces had a certain distinct look where thin letters had more kerning around them than thick letters. Even the really late model typewriters like the above mentioned Selectrics still followed that system. As people got used to seeing monospaced slab serif typefaces, they became specifically associated with typewriters. And as the longevity of Courier proves, they just never went out of style.