First, I will echo complements on aesthetics - I particularly liked how you pulled off the parallax scrolling effect from that diagonal perspective. It makes me think of a stage play or a puppet show. You could potentially lean on that aspect, using elements of said shows to help address the challenge of positioning and perspectives (player/enemies hanging off string, maybe?)
A lot has already been said about positioning, but I will add one thing that I don't think anyone has mentioned yet - I haven't played zaxxon, but in most scrolling shooters when you die, the player typically respawns right away. Maybe I'm sticking to tradition too much, but I feel like this game would benefit just a little bit more in doing that, rather than having the screen fade out and fade back in. The way it is now, I felt as if I've lost all momentum, and have to re-orient myself to what's ahead.
All that aside, I think the game feels good when it comes to movement and shooting, and your music doesn't loop too much nor is too distracting - going to interesting places and knowing when to move on to another chord or melody. I think once the positioning stuff is down, you got a real solid framework for a full game here.