Why Heidegger's stepchildren? The case might be made that Morrison identified himself much more closely with Nietzsche's Dionysos; and that may well be. But I have always also heard echoes of Heidegger (who was also captivated by Nietzsche) in some of Morrison's pieces, like this one. Just listen to the lyrics.
"Riders on the storm." In his June 1933 speech to the Student Association at the university of Heidelberg, Heidegger as the Nazi rector, who regularly ended his speeches of the period with "Heil Hitler," quoted Plato (Republic 497d9) near the end of his speech, assimilating his words to the spirit of National Socialism, with echoes of 18th-century Sturm und drang: "All that is great stands in the storm." This was a romantic notion associated in then contemporary German thinking with the mythos of the Germanic spirit and ancient myths of Wotan and Thor. But here in Morrison, it's just the romance, dark with mystery: Yeah, baby. Maybe Charlie don't surf, but we're gonna ride this hurricane of a storm!
"Into this world we're thrown." Heidegger regularly used the expression "thrown" or "thrownness" (Geworfenheit) to describe Dasein's condition of Being-in-the-World. In Morrison, too, this notion is associated with the raw condition of one's brute existence in the world. The common Existentialist formula was "Existence precedes essence," in the sense that we find ourselves in the world without any fixed antecedent nature or sense of personal identity. The only point, if ever, that one could be described as having a definable nature is after one is dead and gone. Becoming is ended. One's essence becomes fixed in death.
There is a similar obsession with death between the two, as well, which one finds in Heidegger's treatment of Being-toward-death, which he develops famously in connection with an exegesis of Tolstoi's short story, The Death of Ivan Ilyich; as well as throughout Morrison's musical corpus. One sees it briefly here in the "killer on the road" and the reference that "sweet memory will die" if you "give this man a ride," etc.; but elsewhere it appears much more clearly in his repeated declaration "No one here gets out alive," which was both a line in his hit, Five to One, as well as a similarly worded title of his biography, No One Gets Out Alive. In fact, there's death wish written all over many of his lyrics, notably in When the Music's Over: "Cancel my subscription to the resurrection ... Before I sink into the big sleep ... Save us! Jesus! ... When the music's over, turn out the light." We could also reference here the significance of "music" for Dionysus, developed particularly in Nietzsche's The Birth of Tragedy from the Spirit of Music.
In any case, Morrison was a pitifully confused spiritual wreck of a man. Just like the far more 'respectable' Heidegger.