The many-worlds interpretation (MWI) solves the asymmetry in your example.
Many-worlds implies that all possible alternate histories and futures are real, each representing an actual "world" (or "universe").
Symmetry is established since "A observes B" has a symmetric partner world where "B observes A"
I admit I favor the "collapse of the wave function" interpretation, but please feel free to pick your own favorite interpretation here.
Back to the question in the title "Why is there an asymmetry in QM?", I'd like to add the following:
Observing a quantum system (made up of a superposition of eigenstates), collapses the wave function to a single eigenstate of your observation operator and by that destroys superposition. More important: the observer A gathers information about the system B*. The (asymmetric) information flow from B to A drives change in the world...
*: This can be shown, because the information can be erased (by so called quantum erasers) while traveling to the observer A. By that, the superposition state is not affected.