The truse origins of "Graeco-Roman" wrestling
Many modern GR wrestlers and fans (and even some writers) believe that their sport is directly descended from the ancient Greek Olympics. The truth is more complicated.
During the late 19th and early 20th centuries there was a wave of enthusiasm for classical studies throughout Europe, including developments in architecture, literature, the fine arts, dance and - oddly enough - athletics, all inspired by the examples of ancient Greece and Rome. One manifestation of this trend was the birth of the modern Olympic Games, which was originally just one of a number of similar sports festivals combining traditional European folk sports with varying degrees of "Graeco-Roman re-enactment" - see this link for an account of the Graeco-Roman Games in California, 1895.
The French especially enjoyed re-constructions of ancient athletic events as a sort of "sports theater", almost like modern professional wrestling shows, and teams of actors and athletes would rehearse/train for weeks to put on a good performance. Regarding the specific development of the "Greco-Roman" wrestling style, I can do no better than to quote wrestling historian Willie Baxter:
To further clarify, at the time, Baron de Coubertin's selection committee was heavily influenced by the vogue for re-constructions of the ancient Games, and the co-incidence of styles between the native French open-handed wrestling and the styles represented in Classical art were overpowering. Had the French style not resembled what de Coubertin and his colleagues admired in representations of ancient wrestling, Catch-as-catch-can might well have received the Olympic nod, and the history of MMA, Olympic and collegiate wrestling, etc. might have been very different.
When the modern Olympic Games began in Athens in 1896, wrestling was one of the sports competed and like the other events the organisation was amateurish and the standard was very poor. The style of wrestling chosen was Greco-Roman, which despite its grandiose title owes nothing to Greece or Rome but everything to France.
The two Olympic styles are folk styles, which have been greatly technically developed since their Olympic inception, but the senior style, Greco-Roman, one of a family of Mediterranean styles, which have existed in La Provençe (South East France) since antiquity but was first formally codified in 1848 by a lawyer called Innocent Truquettil.
Greco-Roman was initially called in French la luttes à mains platte (open handed wrestling) and technical development began in the ‘athletes’ cafés’ of Lyon and Bordeaux. About 1860 it was sometimes referred to as la lutte Romaine (Roman wrestling) then later as la lutte Grecque (Greek wrestling). Eventually the two names were joined together and it became generally known as Graeco-Roman/Greco-Roman, though as late as 1910 it was frequently referred to in France as la lutte à mains platte.
Pierre de Fridi, Baron de Coubertin (founder of the modern Olympic Games) chose Greco-Roman for the first Olympic Games for several reasons, first because of the name which was chosen because the founders in 1848 genuinely believed that it was the type of wrestling described by Homer in the Iliad (circa 1200 BC). The second reason was because at that time it was the most popular spectator sport in Europe, excluding Britain where Catch-as-catch-can was incredibly popular.