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The Mongols and the Romans only ever were involved on two continents. Alexander really only fought on one, though starting in Europe. A war on only two continents and no oceans is not a ‘world’ war.

The war between Great Britain and France in the late 18th century was possibly the first instance of global warfare, with military land conflict taking part on at least four of the seven continents, and with naval battles on at least four of the five oceans. World War 1 is normally classed as the first world war because it took part on all inhabited continents (i.e. excluding Antartica, though I don’t know what fighting there was in South America) and on all oceans. World War 2 has the same distinction.

That’s not to say that war somewhere around the globe was not continuous. If one is willing to describe inter-tribal conflict involving only a few thousand people as ‘war’ then there has probably never been peace in the world for a few thousand years. Humans are a cautious animal with a natural tendency to imagine danger rather than assume safety. Our natural tendency to be suspicious has kept our ancestors alive despite being relatively slow and weak creatures but it also leads us to attack rather than question whether someone truly is a threat. ‘The others’ are always a threat first, allies and friends later.