You could make a list of the top 50 ways in which EarthBound is unlike any other RPG if you wanted to. One of them is: there are about ten or twelve different random battle tracks, rather than the standard one. They range from mundane battle music (Black Antoids) to goofy (Spiteful Crows, Sharks) to trippy (Frank) and so forth.
But this one is the most "EarthBoundy" (remember that word?) of the bunch, which is fitting since it's the battle music of EarthBound's mascot villains, the Starmen. This music accompanies your fights against all varieties of Starman except the first one, Starman Junior, which gets EarthBound's rare "Super Boss" music for some strange reason.
It's not just the Starmen; it's kind of the "alien machine" theme, shared by some Spinning Robos and Wooly Shamblers and Octobots and such. Giygas doesn't throw a bunch of his alien spawn at you; EarthBound's designers figured, quite reasonably, that an advanced alien lifeform would probably show up with an army of robots.
In EarthBound you start out fighting mostly plants, animals and people that have flipped out thanks to Giygas's evil aura or whatever; but the further you go in the game, the more you have to hack your way through minions directly in Giygas' employ, which include lots of robots. And you hear this music more.