mattkevan's comments are a good primer for quickly identifying one font over another, and are pretty much what we would have done for the "first cut" to disqualify similar faces.
It isn't a job I'd relish today. Back then, in essentially the pre-computer days, you were dealing with thousands of fonts. Today, the choices seem almost infinite.
Firstly, is it serif, sans-serif, handwritten, monospaced etc? This will narrow it down, but distinguishing between similar fonts, such as Arial/Helvetica, Avenir/Futura, Garamond/Caslon etc. requires looking at individual characters.
Lots of characters don't change much between typefaces and can be hard to identify at small sizes, but others can change a lot so they're the ones to look out for - including the lowercase g, uppercase Q and ampersand &.
Is the g double storey or single storey? Open Sans and Noto are identical except for the g (Open Sans is double storey).
What's the tail on the Q like? Does it cross the circle, is it curved or straight? Open Sans and Droid Sans are identical except that the tail of the Q is curved in Droid Sans and straight in Open Sans. (Also the uppercase I has bars in Droid Sans).
Arial and Helvetica can be told apart by looking at the lowercase t. Arial's t terminal is at and angle whereas Helvetica's is horizontal. Also the terminals of the uppercase S are at an angle whereas Helvetica's is horizontal.
Thanks for the answer. I guess I just need more practice.
I recommend "The Elements of Typographic Style" by Robert Bringhurst if you'd like to improve your type identification abilities. The appendices have a good, but by no means exhaustive, reference of some of the better known typefaces.