The word "caliber" has a number of different meanings, but in watchmaking the word is synonymous with "movement." Across the entire industry, one of the few absolutely ubiquitous practices is to call a movement "caliber XYZ123." It's one of those words you can see a million times without wondering why it's used in both the world of firearms and the world of watchmaking, which would seem to have little in common. But, have no fear; Hodinkee's done the wondering (and researching) for you.
The use of "caliber" in the gun world goes back through the French calibre all the way to the late Renaissance Arabic qalib—a mold for casting bullets—and ultimately has its origin in the ancient Greek kalapous, which means a shoemaker's last (the wooden model around which a shoe is constructed).
Interestingly enough, the use of the term in horology is first found, according to the Fondation de la Haute Horlogerie, in the work of the English clockmaker Henry Sully (1680-1729) who worked in France, and used the word "... circa 1715, to denote the layout and dimensions of the different movement pillars, wheels, barrel, etc." The term was used as time went on, to "indicate the shape of the movement, its bridges, the origin of the watch, its maker's name, etc." (i.e. IWC "Jones Caliber") and gradually came to refer to the movement itself. The general connection, then, is that in both watchmaking and gunmaking the term has a history of being used to denote diameter and in both cases, to some extent, has come to refer to a thing whose diameter was specified (maybe by metonymy).
The typical practice nowadays is to refer to a movement as the "caliber xyz123" (that is, "caliber" followed by a movement model number, such as the ETA caliber 2892-A2). Interestingly enough, the word "caliber" has for much of its history also meant a person of a certain standing, and the usage is still alive today; for instance, in the novel The Godfather, the Don's nephew Johnny Fontane refers to the film producer Jack Woltz as "a real .90 caliber pezzonovante [big shot]."
The general standard practice, then, in the French-speaking watchmaking world, was to refer to a movement as a calibre, followed by the diameter of the movement, and then, often, an abbreviation giving the movement's special functions. For example, Jaeger-LeCoultre's manufacturing archives from 1877 (when the company was Manufacture LeCoultre Borgeaud & Cie) list, as one of the movements made, calibre 16T. This tells us that this was a 16 ligne tourbillon movement.
The ligne, by the way, is equal to 2.2558291 millimeters (a 16 ligne movement would therefore be about 36.09mm in diameter). The ligne is still used as a unit of measurement in watchmaking and also in, oddly enough, button making and the measurement of ribbons used for men's hat-bands; it is 1/12 of a pouce (the French inch).
This isn't the only link between watchmaking and the firearms industry, though. In the United States, the so-called American System of watchmaking was one of the first in the world to make extensive use of precision mass production methods and machinery to allow the production of watch movements with interchangeable parts. The system was adapted for watchmaking by American makers like Waltham, from the manufacturing methods used by the Federal armory at Springfield, Massachusetts, where it was used to mass-produce rifles.
Hodinkee's Louis Westphalen notes that in the years leading up to World War II and thereafter, the trend was to shift away from using movement diameter and functions to designate movements as the number of different movements of similar or identical diameters increased. Even before World War I, however, this naming convention wasn't universal; the Valjoux 22 is an example. The practice is still alive here and there, however; Patek Philippe, for instance, has in its catalogue the calibre 17''' LEP PS IRM. This a 17 ligne, lepine, petite seconde, indication réserve de marche movement – or, a 38.35mm Lepine caliber (that is, a movement with bridges for the train wheels and no fusée, named for the watchmaker Jean-Antoine Lepine who developed this movement architecture in the late 18th century) with small seconds and power reserve.