What mechanism is responsible for the β/hFE variations in manufactured bipolar transistors?
beta depends on width of the base region which, along with doping levels within the base, determines the fraction of re-combination within that base region.

thus you have 3 doping levels and temperature as effects.

additionally the Vcb affects the width of collector depletion region which affect the  effective base region; thus current mirrors can seriously mismatch, what with one transistor biased as a diode and the other in normal common-emitter mode