I feel like some of them at least could be singing to us, like Garuda or Titania. Others like Leviathan, Titan, and Ramuh sound like at least part of their songs are sung by their beastmen supporters. Part of the Ramuh song is even in the Slyph language, and the King Moogle Mog fight has each individual moogle stop singing their part of the song when you knock them out. Most of the others are songs seemingly from the primal's point of view about how badass they are or their life story. I'd like to imagine we're hearing that through the echo.

The real answer in most of the cases is probably that Soken just wanted every primal fight to be more interesting than Primal Judgement, it's something cool to listen to in each of the fights, he seems to have fun with it, and he and Koji have even done some of the singing.