The Panzer Dragoon games are indeed a continuing story -- the third game is an RPG after all, but the story is told over multiple games and is slow until the RPG. The main driving story in the games is a reoccurring cycle, individual games are separated by hundreds of years. Long, long ago, society ended in a devastating war which ruined the planet. All old technology was lost, and humanity reduced to nomadic tribes that are on the verge of extinction. Parts of the remaining humans have formed a small empire, which has been digging up ancient artifacts looking for lost technology to rule the world.
It's in this setting that a pair of dragons keep re-emerging over the centuries following the same pattern. A blue dragon chooses a "rider" to bond with, with the goal of stopping a black dragon from reaching an enormous tower. The nature of the black and blue dragons, the purpose of the tower, or why the dragon compels you to stop the black dragon from reaching the tower, are the central mystery to the games that get explored over time. Panzer Dragoon comes first, Panzer Dragoon Zwei is technically a prequel, and Panzer Dragoon Saga is the finale to the story.
You don't need to pay attention to the story in the SHMUP games at all, though. They're told through cutscenes between levels.