You can't really wage anything other than war. You can try, but it sounds rather odd. Indeed the phrase waging war gets stranger the more you look at it. A wage war should be similar to a pay dispute, or what job advertisements so euphemistically refer to as a competitive salary.
There is a connection between all these different wages, and indeed to wagers. But you have to go back to the fourteenth century.
A wage was, originally, a pledge or deposit. It was something given in security. From this wage you quite easily get to the modern wager: it is merely the stake, or deposit, thrown down by a gambler. It's also reasonably simple to see how money given in security could end up meaning money given as pay. And war? That's a trifle stranger. It involves trial by combat.
In medieval law it was considered quite reasonable to settle a legal dispute by duelling to the death. Though somebody had to die, lawyers fees were, at least, kept to a minimum. In Latin that was vadiare duellum; in English you waged [pledged yourself to] battle.
Not war. Battle. It was, after all, a technical legal term for the violent and slicing resolution of individual arguments. You wagered your body in mortal combat. However, it's easy to see how the sense of waging battle extended from the promise of violence to the act, and then expanded from the battle to the war.
This last shift in meaning could reasonably be described as wage inflation.
The Inky Fool solving a wage dispute