I've only played the GBA/DS versions.
I just booted this game and all I see are "2-4 Player" modes? Isn't there a regular story/campaign mode? Or by 2 players it means Player 1 and Computer?
The two 2P modes, if played against Computer, are sort of campaigns. You need to clear all 15 maps, after which a final "boss map" is unlocked. There are two final maps per mode depending on which army you play. When the boss map is cleared you get a cute little ending. There's nothing in terms of story, though.
Note that if playing with "Domination" enabled the final maps are really tough, since those start off with the AI having lots of units deployed, and lots of buildings already captured.
The "Classic" 2P campaign consists of all the maps from the Famicom Wars (ie. the "NES" version of the game). The "New" 2P campaign has new maps, with some added features not present in Famicom Wars: Labs that grant you a Prototype Tank when captured, maps with railroads/depots/trains, and if I'm remembering correctly Forts are also only featured in the "New" maps.