Why don't flyback diodes in H-bridge damage power supply?
The diodes serve two distinct purposes.


Under regenerative braking, they return the generated voltage to the power supply (where with suitable electronics, it can be used to recharge the battery). Note that unless the motor is being run above its normal speed, the generated voltage will be no more than the supply voltage, so it is within the voltage rating of the power supply. So the power supply can normally withstand this - but if it can't absorb current (either to charge a battery, or dump it into a braking resistor) there will be little or no braking effect.
The diodes also return inductive spikes (from the motor brushes) to the supply, and these may be hundreds of volts, for a very short duration, which can prove destructive to power supplies. Then to answer the actual question - the supply CAN be damaged by high voltage spikes, so its designer must take precautions to prevent that damage - like an inductor (ferrite bead) in series, and ample decoupling capacitors across the supply, and possibly a transient suppressor or varistor to absorb HV transients


Note there isn't generally enough energy in these spikes to do any damage whatsoever to a primary cell, so relax if you're connecting the bridge straight to a battery. But regulated supplies that aren't designed to drive motors may be a problem.