Every political campaign is a march across a field. The word comes from French campagne, from Italian campagna, from Latin campus — 'an open plain'. A military campaign was the period when armies left their winter quarters and fought in open country. The field was the campaign.
Latin campus is one of the most productive roots in English. A camp is a field settlement. A campus is a field for learning (borrowed from Princeton's use in the 1770s). Champion came through Old French from Latin campiō, 'a fighter in the field'. Even Champagne, the French wine region, takes its name
The political sense appeared in Britain around 1720, when writers began comparing electioneering to military operations. The metaphor stuck because it works: a campaign requires strategy, logistics, territory to be won, and an opponent to defeat.
The advertising sense followed the same military logic. A marketing campaign is an organised offensive — a sustained effort across multiple fronts. Every use of the word, from warfare to elections to product launches, preserves the image of troops marching across an open Latin field.