One urge common to wargamers but rare among other board gaming flavors (although shared with RPGs) is to seek out and play the "epic," "campaign" or "monster" game. All three are just different manifestations of the same desire, I think, to capture the sweep and drama of a bigger-than-life gaming experience.
Warfare, by its very nature, lends itself to the grand sweep of things and its interrelated parts are well-suited to being linked. Even the otherwise-rather basic Commands and Colors system includes "Epic" and "Overlord" versions. Monster games requiring a half-dozen players achieve the links laterally.
But perhaps the most common way to create a campaign narrative is by linking a series of individual scenarios together into a longer metagame. The whole idea of "scenarios" is itself peculiar to wargames. I don't think there's anything like it in other boardgames.
The original Squad Leader included a way to build a campaign narrative including the opportunity for promoting a "personal leader" but there were many ways to link games even in ASL. The historical modules like Red Barricades and Kampfgruppe Peiper have obvious linkages, for example, but others are possible.
Myself and my regular ASL partners engaged in a giant metacampaign for the 50th anniversary of World War II playing the "official" Avalon Hill ASL scenarios in chronological order more or less on the 50th anniversaries of the actual fights. Like perhaps most such overly ambitious wargame projects it didn't reach completion (at least not yet) but it did reach fruition. We made it well in the summer of 1944 before life changes knocked the effort off schedule. Finishing the last year of the war is an ongoing, if very occasional, project to this day, even as the 60th anniversaries have now passed.
Still, we got some 200+ games played in some kind of order and had a good time doing so. There's a unique satisfaction to playing on the epic scale.