Do academic seasons matter post-PhD?
This is based on my experience in mathematics in the United States (other cases may differ):

In principle, a postdoctoral position could start at any time if grant funding is available.  In practice, it is generally tied to the academic calendar the same way other positions are, and any deviations from that schedule would be unusual.  If you are doing an advertised search for a candidate, then there are enormous advantages to timing it with the usual job market cycle (at other times of year very few applicants will be available), so off-season hiring typically takes place only when a perfect match is made through personal connections.  The other issue is that most Ph.D. students get their degrees at the end of the spring semester, and most faculty positions start in the fall semester, so academic-year-based postdocs are convenient for most applicants.