The word "company" actually derives from the Old French word "compaignie," which itself comes from the Latin word "compagnia," meaning "association," "fellowship," or "group." This broader sense of togetherness is what the word "company" conveys in modern English.
While sharing wine and bread was certainly a common practice in Roman times, and likely contributed to the formation of social groups, it's important to remember that "company" encompasses a wider range of groups and activities beyond just shared meals.